2011
2011
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Paper 1, Section I,
2011 comment(a) State, without proof, the Bolzano-Weierstrass Theorem.
(b) Give an example of a sequence that does not have a convergent subsequence.
(c) Give an example of an unbounded sequence having a convergent subsequence.
(d) Let , where denotes the integer part of . Find all values such that the sequence has a subsequence converging to . For each such value, provide a subsequence converging to it.
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Paper 1, Section I, D
2011 commentFind the radius of convergence of each of the following power series. (i) (ii)
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Paper 1, Section II, F
2011 comment(a) State, without proof, the ratio test for the series , where . Give examples, without proof, to show that, when and , the series may converge or diverge.
(b) Prove that .
(c) Now suppose that and that, for large enough, where . Prove that the series converges.
[You may find it helpful to prove the inequality for .]
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Paper 1, Section II, E
2011 commentState and prove the Intermediate Value Theorem.
A fixed point of a function is an with . Prove that every continuous function has a fixed point.
Answer the following questions with justification.
(i) Does every continuous function have a fixed point?
(ii) Does every continuous function have a fixed point?
(iii) Does every function (not necessarily continuous) have a fixed point?
(iv) Let be a continuous function with and . Can have exactly two fixed points?
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Paper 1, Section II, E
2011 commentFor each of the following two functions , determine the set of points at which is continuous, and also the set of points at which is differentiable.
By modifying the function in (i), or otherwise, find a function (not necessarily continuous) such that is differentiable at 0 and nowhere else.
Find a continuous function such that is not differentiable at the points , but is differentiable at all other points.
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Paper 1, Section II, D
2011 commentState and prove the Fundamental Theorem of Calculus.
Let be integrable, and set for . Must be differentiable?
Let be differentiable, and set for . Must the Riemann integral of from 0 to 1 exist?
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Paper 2, Section I, A
2011 comment(a) Consider the homogeneous th-order difference equation
where the coefficients are constants. Show that for the sequence is a solution if and only if , where
State the general solution of if and for some constant .
(b) Find an inhomogeneous difference equation that has the general solution
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Paper 2, Section I,
2011 comment(a) For a differential equation of the form , explain how can be used to determine the stability of any equilibrium solutions and justify your answer.
(b) Find the equilibrium solutions of the differential equation
and determine their stability. Sketch representative solution curves in the -plane.
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Paper 2, Section II, A
2011 comment(a) Find the general real solution of the system of first-order differential equations
where is a real constant.
(b) Find the fixed points of the non-linear system of first-order differential equations
and determine their nature. Sketch the phase portrait indicating the direction of motion along trajectories.
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Paper 2, Section II, A
2011 comment(a) A surface in is defined by the equation , where is a constant. Show that the partial derivatives on this surface satisfy
(b) Now let , where is a constant.
(i) Find expressions for the three partial derivatives and on the surface , and verify the identity .
(ii) Find the rate of change of in the radial direction at the point .
(iii) Find and classify the stationary points of .
(iv) Sketch contour plots of in the -plane for the cases and .
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Paper 2, Section II,
2011 comment(a) Define the Wronskian of two solutions and of the differential equation
and state a necessary and sufficient condition for and to be linearly independent. Show that satisfies the differential equation
(b) By evaluating the Wronskian, or otherwise, find functions and such that has solutions and . What is the value of Is there a unique solution to the differential equation for with initial conditions ? Why or why not?
(c) Write down a third-order differential equation with constant coefficients, such that and are both solutions. Is the solution to this equation for with initial conditions unique? Why or why not?
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Paper 2, Section II, A
2011 comment(a) The circumference of an ellipse with semi-axes 1 and is given by
Setting , find the first three terms in a series expansion of around .
(b) Euler proved that also satisfies the differential equation
Use the substitution for to find a differential equation for , where . Show that this differential equation has regular singular points at and .
Show that the indicial equation at has a repeated root, and find the recurrence relation for the coefficients of the corresponding power-series solution. State the form of a second, independent solution.
Verify that the power-series solution is consistent with your answer in (a).
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Paper 4, Section I, B
2011 commentThe motion of a planet in the gravitational field of a star of mass obeys
where and are polar coordinates in a plane and is a constant. Explain one of Kepler's Laws by giving a geometrical interpretation of .
Show that circular orbits are possible, and derive another of Kepler's Laws relating the radius and the period of such an orbit. Show that any circular orbit is stable under small perturbations that leave unchanged.
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Paper 4, Section I, B
2011 commentInertial frames and in two-dimensional space-time have coordinates and , respectively. These coordinates are related by a Lorentz transformation with the velocity of relative to . Show that if and then the Lorentz transformation can be expressed in the form
Deduce that .
Use the form to verify that successive Lorentz transformations with velocities and result in another Lorentz transformation with velocity , to be determined in terms of and .
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Paper 4, Section II, B
2011 commentA particle with mass and position is subject to a force
(a) Suppose that . Show that
is constant, and interpret this result, explaining why the field plays no role.
(b) Suppose, in addition, that and that both and depend only on . Show that
is independent of time if , for any constant .
(c) Now specialise further to the case . Explain why the result in (b) implies that the motion of the particle is confined to a plane. Show also that
is constant provided takes a certain form, to be determined.
[ Recall that and that if depends only on then
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Paper 4, Section II, B
2011 commentThe trajectory of a particle is observed in a frame which rotates with constant angular velocity relative to an inertial frame . Given that the time derivative in of any vector is
where a dot denotes a time derivative in , show that
where is the force on the particle and is its mass.
Let be the frame that rotates with the Earth. Assume that the Earth is a sphere of radius . Let be a point on its surface at latitude , and define vertical to be the direction normal to the Earth's surface at .
(a) A particle at is released from rest in and is acted on only by gravity. Show that its initial acceleration makes an angle with the vertical of approximately
working to lowest non-trivial order in .
(b) Now consider a particle fired vertically upwards from with speed . Assuming that terms of order and higher can be neglected, show that it falls back to Earth under gravity at a distance
from . [You may neglect the curvature of the Earth's surface and the vertical variation of gravity.]
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Paper 4, Section II, B
2011 commentA rocket carries equipment to collect samples from a stationary cloud of cosmic dust. The rocket moves in a straight line, burning fuel and ejecting gas at constant speed relative to itself. Let be the speed of the rocket, its total mass, including fuel and any dust collected, and the total mass of gas that has been ejected. Show that
assuming that all external forces are negligible.
(a) If no dust is collected and the rocket starts from rest with mass , deduce that
(b) If cosmic dust is collected at a constant rate of units of mass per unit time and fuel is consumed at a constant rate , show that, with the same initial conditions as in (a),
Verify that the solution in (a) is recovered in the limit .
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Paper 4, Section II, B
2011 comment(a) Write down the relativistic energy of a particle of rest mass and speed . Find the approximate form for when is small compared to , keeping all terms up to order . What new physical idea (when compared to Newtonian Dynamics) is revealed in this approximation?
(b) A particle of rest mass is fired at an identical particle which is at rest in the laboratory frame. Let be the relativistic energy and the speed of the incident particle in this frame. After the collision, there are particles in total, each with rest mass . Assuming that four-momentum is conserved, find a lower bound on and hence show that
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Paper 3, Section I, D
2011 comment(a) Let be the group of symmetries of the cube, and consider the action of on the set of edges of the cube. Determine the stabilizer of an edge and its orbit. Hence compute the order of .
(b) The symmetric group acts on the set , and hence acts on by . Determine the orbits of on .
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Paper 3, Section I, D
2011 commentState and prove Lagrange's Theorem.
Show that the dihedral group of order has a subgroup of order for every dividing .
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Paper 3, Section II, D
2011 comment(a) Let be a finite group, and let . Define the order of and show it is finite. Show that if is conjugate to , then and have the same order.
(b) Show that every can be written as a product of disjoint cycles. For , describe the order of in terms of the cycle decomposition of .
(c) Define the alternating group . What is the condition on the cycle decomposition of that characterises when ?
(d) Show that, for every has a subgroup isomorphic to .
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Paper 3, Section II, D
2011 comment(a) Let
and, for a prime , let
where consists of the elements , with addition and multiplication mod .
Show that and are groups under matrix multiplication.
[You may assume that matrix multiplication is associative, and that the determinant of a product equals the product of the determinants.]
By defining a suitable homomorphism from , show that
is a normal subgroup of .
(b) Define the group , and show that it has order 480 . By defining a suitable homomorphism from to another group, which should be specified, show that the order of is 120 .
Find a subgroup of of index 2 .
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Paper 3, Section II,
2011 comment(a) State the orbit-stabilizer theorem.
Let a group act on itself by conjugation. Define the centre of , and show that consists of the orbits of size 1 . Show that is a normal subgroup of .
(b) Now let , where is a prime and . Show that if acts on a set , and is an orbit of this action, then either or divides .
Show that .
By considering the set of elements of that commute with a fixed element not in , show that cannot have order .
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Paper 3, Section II, D
2011 comment(a) Let be a finite group and let be a subgroup of . Show that if then is normal in .
Show that the dihedral group of order has a normal subgroup different from both and .
For each integer , give an example of a finite group , and a subgroup , such that and is not normal in .
(b) Show that is a simple group.
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Paper 4, Section I, E
2011 commentWhat does it mean to say that a function has an inverse? Show that a function has an inverse if and only if it is a bijection.
Let and be functions from a set to itself. Which of the following are always true, and which can be false? Give proofs or counterexamples as appropriate.
(i) If and are bijections then is a bijection.
(ii) If is a bijection then and are bijections.
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Paper 4, Section I, E
2011 commentWhat is an equivalence relation on a set If is an equivalence relation on , what is an equivalence class of ? Prove that the equivalence classes of form a partition of .
Let be the relation on the positive integers defined by if either divides or divides . Is an equivalence relation? Justify your answer.
Write down an equivalence relation on the positive integers that has exactly four equivalence classes, of which two are infinite and two are finite.
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Paper 4, Section II, E
2011 comment(a) What is the highest common factor of two positive integers and ? Show that the highest common factor may always be expressed in the form , where and are integers.
Which positive integers have the property that, for any positive integers and , if divides then divides or divides ? Justify your answer.
Let be distinct prime numbers. Explain carefully why cannot equal .
[No form of the Fundamental Theorem of Arithmetic may be assumed without proof.]
(b) Now let be the set of positive integers that are congruent to 1 mod 10 . We say that is irreducible if and whenever satisfy then or . Do there exist distinct irreducibles with
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Paper 4, Section II, E
2011 commentState Fermat's Theorem and Wilson's Theorem.
Let be a prime.
(a) Show that if then the equation has no solution.
(b) By considering !, or otherwise, show that if then the equation does have a solution.
(c) Show that if then the equation has no solution other than .
(d) Using the fact that , find a solution of that is not .
[Hint: how are the complex numbers and related?]
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Paper 4, Section II,
2011 commentDefine the binomial coefficient , where is a positive integer and is an integer with . Arguing from your definition, show that .
Prove the binomial theorem, that for any real number .
By differentiating this expression, or otherwise, evaluate and . By considering the identity , or otherwise, show that
Show that
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Paper 4, Section II, E
2011 commentShow that, for any set , there is no surjection from to the power-set of .
Show that there exists an injection from to .
Let be a subset of . A section of is a subset of of the form
where and with . Prove that there does not exist a set such that every set is a section of .
Does there exist a set such that every countable set is a section of [There is no requirement that every section of should be countable.] Justify your answer.
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Paper 2, Section I, F
2011 commentLet be a random variable taking non-negative integer values and let be a random variable taking real values.
(a) Define the probability-generating function . Calculate it explicitly for a Poisson random variable with mean .
(b) Define the moment-generating function . Calculate it explicitly for a normal random variable .
(c) By considering a random sum of independent copies of , prove that, for general and is the moment-generating function of some random variable.
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Paper 2, Section I, F
2011 commentWhat does it mean to say that events are (i) pairwise independent, (ii) independent?
Consider pairwise disjoint events and , with
Let . Prove that the events and are pairwise independent if and only if
Prove or disprove that there exist and such that these three events are independent.
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Paper 2, Section II, F
2011 comment(a) Let be pairwise disjoint events such that their union gives the whole set of outcomes, with for . Prove that for any event with and for any
(b) A prince is equally likely to sleep on any number of mattresses from six to eight; on half the nights a pea is placed beneath the lowest mattress. With only six mattresses his sleep is always disturbed by the presence of a pea; with seven a pea, if present, is unnoticed in one night out of five; and with eight his sleep is undisturbed despite an offending pea in two nights out of five.
What is the probability that, on a given night, the prince's sleep was undisturbed?
On the morning of his wedding day, he announces that he has just spent the most peaceful and undisturbed of nights. What is the expected number of mattresses on which he slept the previous night?
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Paper 2, Section II, F
2011 comment(a) State Markov's inequality.
(b) Let be a given positive integer. You toss an unbiased coin repeatedly until the first head appears, which occurs on the th toss. Next, I toss the same coin until I get my first tail, which occurs on my th toss. Then you continue until you get your second head with a further tosses; then I continue with a further tosses until my second tail. We continue for turns like this, and generate a sequence , of random variables. The total number of tosses made is . (For example, for , a sequence of outcomes gives and .)
Find the probability-generating functions of the random variables and . Hence or otherwise obtain the mean values and .
Obtain the probability-generating function of the random variable , and find the mean value .
Prove that, for ,
For , calculate , and confirm that it satisfies Markov's inequality.
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Paper 2, Section II, F
2011 commentI was given a clockwork orange for my birthday. Initially, I place it at the centre of my dining table, which happens to be exactly 20 units long. One minute after I place it on the table it moves one unit towards the left end of the table or one unit towards the right, each with probability 1/2. It continues in this manner at one minute intervals, with the direction of each move being independent of what has gone before, until it reaches either end of the table where it promptly falls off. If it falls off the left end it will break my Ming vase. If it falls off the right end it will land in a bucket of sand leaving the vase intact.
(a) Derive the difference equation for the probability that the Ming vase will survive, in terms of the current distance from the orange to the left end, where .
(b) Derive the corresponding difference equation for the expected time when the orange falls off the table.
(c) Write down the general formula for the solution of each of the difference equations from (a) and (b). [No proof is required.]
(d) Based on parts (a)-(c), calculate the probability that the Ming vase will survive if, instead of placing the orange at the centre of the table, I place it initially 3 units from the right end of the table. Calculate the expected time until the orange falls off.
(e) Suppose I place the orange 3 units from the left end of the table. Calculate the probability that the orange will fall off the right end before it reaches a distance 1 unit from the left end of the table.
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Paper 2, Section II, F
2011 commentA circular island has a volcano at its central point. During an eruption, lava flows from the mouth of the volcano and covers a sector with random angle (measured in radians), whose line of symmetry makes a random angle with some fixed compass bearing.
The variables and are independent. The probability density function of is constant on and the probability density function of is of the form where , and is a constant.
(a) Find the value of . Calculate the expected value and the variance of the sector angle . Explain briefly how you would simulate the random variable using a uniformly distributed random variable .
(b) and are two houses on the island which are collinear with the mouth of the volcano, but on different sides of it. Find
(i) the probability that is hit by the lava;
(ii) the probability that both and are hit by the lava;
(iii) the probability that is not hit by the lava given that is hit.
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Paper 3, Section I, C
2011 commentCartesian coordinates and spherical polar coordinates are related by
Find scalars and unit vectors such that
Verify that the unit vectors are mutually orthogonal.
Hence calculate the area of the open surface defined by , , where and are constants.
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Paper 3, Section I, C
2011 commentState the value of and find , where .
Vector fields and in are given by and , where is a constant and is a constant vector. Calculate the second-rank tensor , and deduce that and . When , show that and
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Paper 3, Section II, C
2011 commentWrite down the most general isotropic tensors of rank 2 and 3. Use the tensor transformation law to show that they are, indeed, isotropic.
Let be the sphere . Explain briefly why
is an isotropic tensor for any . Hence show that
for some scalars and , which should be determined using suitable contractions of the indices or otherwise. Deduce the value of
where is a constant vector.
[You may assume that the most general isotropic tensor of rank 4 is
where and are scalars.]
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Paper 3, Section II, C
2011 commentState the divergence theorem for a vector field in a region bounded by a piecewise smooth surface with outward normal .
Show, by suitable choice of , that
for a scalar field .
Let be the paraboloidal region given by and , where and are positive constants. Verify that holds for the scalar field .
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Paper 3, Section II, C
2011 commentThe electric field due to a static charge distribution with density satisfies
where is the corresponding electrostatic potential and is a constant.
(a) Show that the total charge contained within a closed surface is given by Gauss' Law
Assuming spherical symmetry, deduce the electric field and potential due to a point charge at the origin i.e. for .
(b) Let and , with potentials and respectively, be the solutions to (1) arising from two different charge distributions with densities and . Show that
for any region with boundary , where points out of .
(c) Suppose that for and that , a constant, on . Use the results of (a) and (b) to show that
[You may assume that as sufficiently rapidly that any integrals over the 'sphere at infinity' in (2) are zero.]
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Paper 3, Section II, C
2011 commentThe vector fields and obey the evolution equations
where is a given vector field and is a given scalar field. Use suffix notation to show that the scalar field obeys an evolution equation of the form
where the scalar field should be identified.
Suppose that and . Show that, if on the surface of a fixed volume with outward normal , then
Suppose that with respect to spherical polar coordinates, and that . Show that
and calculate the value of when is the sphere .
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Paper 1, Section I,
2011 commentFor define the principal value of and hence of . Hence find all solutions to (i) (ii) ,
and sketch the curve .
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Paper 1, Section I, A
2011 commentThe matrix
represents a linear map with respect to the bases
Find the matrix that represents with respect to the bases
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Paper 1, Section II,
2011 commentExplain why each of the equations
describes a straight line, where and are constant vectors in and are non-zero, and is a real parameter. Describe the geometrical relationship of a, and to the relevant line, assuming that .
Show that the solutions of (2) satisfy an equation of the form (1), defining and in terms of and such that and . Deduce that the conditions on and are sufficient for (2) to have solutions.
For each of the lines described by (1) and (2), find the point that is closest to a given fixed point .
Find the line of intersection of the two planes and , where and are constant unit vectors, and and are constants. Express your answer in each of the forms (1) and (2), giving both and as linear combinations of and .
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Paper 1, Section II,
2011 commentThe map is defined for , where is a unit vector in and is a constant.
(a) Find the inverse map , when it exists, and determine the values of for which it does.
(b) When is not invertible, find its image and kernel, and explain geometrically why these subspaces are perpendicular.
(c) Let . Find the components of the matrix such that . When is invertible, find the components of the matrix such that .
(d) Now let be as defined in (c) for the case , and let
By analysing a suitable determinant, for all values of find all vectors such that . Explain your results by interpreting and geometrically.
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Paper 1, Section II, B
2011 comment(a) Find the eigenvalues and eigenvectors of the matrix
(b) Under what conditions on the matrix and the vector in does the equation
have 0,1 , or infinitely many solutions for the vector in ? Give clear, concise arguments to support your answer, explaining why just these three possibilities are allowed.
(c) Using the results of , or otherwise, find all solutions to when
in each of the cases .
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Paper 1, Section II, B
2011 comment(a) Let be a real symmetric matrix. Prove the following.
(i) Each eigenvalue of is real.
(ii) Each eigenvector can be chosen to be real.
(iii) Eigenvectors with different eigenvalues are orthogonal.
(b) Let be a real antisymmetric matrix. Prove that each eigenvalue of is real and is less than or equal to zero.
If and are distinct, non-zero eigenvalues of , show that there exist orthonormal vectors with
Part IA, 2011 List of Questions
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, Complex Analysis
2011 commentComplex Analysis or Complex Methods
Complex Methods
Electromagnetism
Fluid Dynamics
Geometry
Groups, Rings and Modules
Linear Algebra
Markov Chains
Methods
Metric and Topological Spaces
Numerical Analysis
Optimization
Quantum Mechanics
Statistics
Variational Principles
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Paper 2, Section I, E
2011 commentDefine differentiability of a function . Let be a constant. For which points is
differentiable? Justify your answer.
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Paper 3, Section I,
2011 commentSuppose is a uniformly continuous mapping from a metric space to a metric space . Prove that is a Cauchy sequence in for every Cauchy sequence in .
Let be a continuous mapping between metric spaces and suppose that has the property that is a Cauchy sequence whenever is a Cauchy sequence. Is it true that must be uniformly continuous? Justify your answer.
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Paper 4, Section I, E
2011 commentLet denote the set of bounded real-valued functions on . A distance on is defined by
Given that is a metric space, show that it is complete. Show that the subset of continuous functions is a closed set.
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Paper 1, Section II, E
2011 commentWhat is meant by saying that a sequence of functions converges uniformly to a function ?
Let be a sequence of differentiable functions on with continuous and such that converges for some point . Assume in addition that converges uniformly on . Prove that converges uniformly to a differentiable function on and for all . [You may assume that the uniform limit of continuous functions is continuous.]
Show that the series
converges for and is uniformly convergent on for any . Show that is differentiable on and
[You may use the Weierstrass -test provided it is clearly stated.]
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Paper 2, Section II, E
2011 commentWhat is meant by saying that two norms on a real vector space are Lipschitz equivalent?
Show that any two norms on are Lipschitz equivalent. [You may assume that a continuous function on a closed bounded set in has closed bounded image.]
Show that defines a norm on the space of continuous real-valued functions on . Is it Lipschitz equivalent to the uniform norm? Justify your answer. Prove that the normed space is not complete.
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Paper 3, Section II, E
2011 commentConsider a map .
Assume is differentiable at and let denote the derivative of at . Show that
for any .
Assume now that is such that for some fixed and for every the limit
exists. Is it true that is differentiable at Justify your answer.
Let denote the set of all real matrices which is identified with . Consider the function given by . Explain why is differentiable. Show that the derivative of at the matrix is given by
for any matrix . State carefully the inverse function theorem and use it to prove that there exist open sets and containing the identity matrix such that given there exists a unique such that .
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Paper 4, Section II, E
2011 commentDefine a contraction mapping and state the contraction mapping theorem.
Let be a non-empty complete metric space and let be a map. Set and . Assume that for some integer is a contraction mapping. Show that has a unique fixed point and that any has the property that as .
Let be the set of continuous real-valued functions on with the uniform norm. Suppose is defined by
for all and . Show that is not a contraction mapping but that is.
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Paper 4, Section I, E
2011 commentLet be an analytic function in an open subset of the complex plane. Prove that has derivatives of all orders at any point in . [You may assume Cauchy's integral formula provided it is clearly stated.]
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Paper 3, Section II, E
2011 commentLet be a continuous function such that
for any closed curve which is the boundary of a rectangle in with sides parallel to the real and imaginary axes. Prove that is analytic.
Let be continuous. Suppose in addition that is analytic at every point with non-zero imaginary part. Show that is analytic at every point in
Let be the upper half-plane of complex numbers with positive imaginary part . Consider a continuous function such that is analytic on and . Define by
Show that is analytic.
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Paper 1, Section I, A
2011 commentDerive the Cauchy-Riemann equations satisfied by the real and imaginary parts of a complex analytic function .
If is constant on , prove that is constant on .
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Paper 1, Section II, A
2011 comment(i) Let and let
Here the logarithms take their principal values. Give a sketch to indicate the positions of the branch cuts implied by the definitions of and .
(ii) Let . Explain why is analytic in the annulus for any . Obtain the first three terms of the Laurent expansion for around in this annulus and hence evaluate
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Paper 2, Section II, A
2011 comment(i) Let be an anticlockwise contour defined by a square with vertices at where
for large integer . Let
Assuming that as , prove that, if is not an integer, then
(ii) Deduce the value of
(iii) Briefly justify the assumption that as .
[Hint: For part (iii) it is sufficient to consider, at most, one vertical side of the square and one horizontal side and to use a symmetry argument for the remaining sides.]
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Paper 3, Section I, D
2011 commentWrite down the function that satisfies
The circular arcs and in the complex -plane are defined by
respectively. You may assume without proof that the mapping from the complex -plane to the complex -plane defined by
takes to the line and to the line , where , and that the region in the -plane exterior to both the circles and maps to the region in the -plane given by .
Use the above mapping to solve the problem
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Paper 4, Section II, D
2011 commentState and prove the convolution theorem for Laplace transforms.
Use Laplace transforms to solve
with , where is the Heaviside function. You may assume that the Laplace transform, , of exists for Re sufficiently large.
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Paper 2, Section I,
2011 commentMaxwell's equations are
Find the equation relating and that must be satisfied for consistency, and give the interpretation of this equation.
Now consider the "magnetic limit" where and the term is neglected. Let be a vector potential satisfying the gauge condition , and assume the scalar potential vanishes. Find expressions for and in terms of and show that Maxwell's equations are all satisfied provided satisfies the appropriate Poisson equation.
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Paper 4, Section I, C
2011 commentA plane electromagnetic wave in a vacuum has electric field
What are the wavevector, polarization vector and speed of the wave? Using Maxwell's equations, find the magnetic field B. Assuming the scalar potential vanishes, find a possible vector potential for this wave, and verify that it gives the correct and .
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Paper 1, Section II, D
2011 commentStarting from the relevant Maxwell equation, derive Gauss's law in integral form.
Use Gauss's law to obtain the potential at a distance from an infinite straight wire with charge per unit length.
Write down the potential due to two infinite wires parallel to the -axis, one at with charge per unit length and the other at with charge per unit length.
Find the potential and the electric field in the limit with where is fixed. Sketch the equipotentials and the electric field lines.
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Paper 2, Section II, C
2011 comment(i) Consider an infinitely long solenoid parallel to the -axis whose cross section is a simple closed curve of arbitrary shape. A current , per unit length of the solenoid, flows around the solenoid parallel to the plane. Show using the relevant Maxwell equation that the magnetic field inside the solenoid is uniform, and calculate its magnitude.
(ii) A wire loop in the shape of a regular hexagon of side length carries a current . Use the Biot-Savart law to calculate at the centre of the loop.
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Paper 3, Section II, C
2011 commentShow, using the vacuum Maxwell equations, that for any volume with surface ,
What is the interpretation of this equation?
A uniform straight wire, with a circular cross section of radius , has conductivity and carries a current . Calculate at the surface of the wire, and hence find the flux of into unit length of the wire. Relate your result to the resistance of the wire, and the rate of energy dissipation.
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Paper 1, Section I, B
2011 commentInviscid fluid is contained in a square vessel with sides of length lying between . The base of the container is at where and the horizontal surface is at when the fluid is at rest. The variation of pressure of the air above the fluid may be neglected.
Small amplitude surface waves are excited in the vessel.
(i) Now let . Explain why on dimensional grounds the frequencies of such waves are of the form
for some positive dimensionless constants , where is the gravitational acceleration.
It is given that the velocity potential is of the form
where and are integers and is a constant.
(ii) Why do cosines, rather than sines, appear in this expression?
(iii) Give an expression for in terms of and .
(iv) Give all possible values that can take between 1 and 10 inclusive. How many different solutions for correspond to each of these values of
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Paper 2, Section I, D
2011 commentA body of volume lies totally submerged in a motionless fluid of uniform density . Show that the force on the body is given by
where is the pressure in the fluid and is atmospheric pressure. You may use without proof the generalised divergence theorem in the form
Deduce that
where is the vertically upward unit vector. Interpret this result.
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Paper 1, Section II, B
2011 commentA spherical bubble in an incompressible fluid of density has radius . Write down an expression for the velocity field at a radius .
The pressure far from the bubble is . What is the pressure at radius ?
Find conditions on and its time derivatives that ensure that the maximum pressure in the fluid is reached at a radius where . Give an expression for this maximum pressure when the conditions hold.
Give the most general form of that ensures that the pressure at is for all time.
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Paper 3, Section II,
2011 commentWater of constant density flows steadily through a long cylindrical tube, the wall of which is elastic. The exterior radius of the tube at a distance along the tube, , is determined by the pressure in the tube, , according to
where and are the radius and pressure far upstream , and is a positive constant.
The interior radius of the tube is , where , the thickness of the wall, is a given slowly varying function of which is zero at both ends of the pipe. The velocity of the water in the pipe is and the water enters the pipe at velocity .
Show that satisfies
where and . Sketch the graph of against .
Let be the maximum value of in the tube. Show that the flow is only possible if does not exceed a certain critical value . Find in terms of .
Show that, under conditions to be determined (which include a condition on the value of , the water can leave the pipe with speed less than .
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Paper 4, Section II, D
2011 commentShow that an irrotational incompressible flow can be determined from a velocity potential that satisfies .
Given that the general solution of in plane polar coordinates is
obtain the corresponding fluid velocity.
A two-dimensional irrotational incompressible fluid flows past the circular disc with boundary . For large , the flow is uniform and parallel to the -axis . Write down the boundary conditions for large and on , and hence derive the velocity potential in the form
where is the circulation.
Show that the acceleration of the fluid at and is
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Paper 1, Section I, F
2011 commentSuppose that is the upper half-plane, . Using the Riemannian metric , define the length of a curve and the area of a region in .
Find the area of
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Paper 3, Section I, F
2011 commentLet denote anti-clockwise rotation of the Euclidean plane through an angle about a point .
Show that is a composite of two reflexions.
Assume . Show that the composite is also a rotation . Find and .
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Paper 2, Section II, F
2011 commentSuppose that is stereographic projection. Show that, via , every rotation of corresponds to a Möbius transformation in .
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Paper 3, Section II, F
2011 commentSuppose that is a unit speed curve in . Show that the corresponding surface of revolution obtained by rotating this curve about the -axis has Gaussian curvature .
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Paper 4, Section II, F
2011 commentSuppose that is a point on a Riemannian surface . Explain the notion of geodesic polar co-ordinates on in a neighbourhood of , and prove that if is a geodesic circle centred at of small positive radius, then the geodesics through meet at right angles.
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Paper 2, Section I, F
2011 commentShow that the quaternion group , with , , is not isomorphic to the symmetry group of the square.
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Paper 3, Section I,
2011 commentSuppose that is an integral domain containing a field and that is finitedimensional as a -vector space. Prove that is a field.
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Paper 4, Section I, F
2011 commentA ring satisfies the descending chain condition (DCC) on ideals if, for every sequence of ideals in , there exists with Show that does not satisfy the DCC on ideals.
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Paper 1, Section II, F
2011 comment(i) Suppose that is a finite group of order , where is prime and does not divide . Prove the first Sylow theorem, that has at least one subgroup of order , and state the remaining Sylow theorems without proof.
(ii) Suppose that are distinct primes. Show that there is no simple group of order .
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Paper 2, Section II, F
2011 commentDefine the notion of a Euclidean domain and show that is Euclidean.
Is prime in ?
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Paper 3, Section II, F
2011 commentSuppose that is a matrix over . What does it mean to say that can be brought to Smith normal form?
Show that the structure theorem for finitely generated modules over (which you should state) follows from the existence of Smith normal forms for matrices over .
Bring the matrix to Smith normal form.
Suppose that is the -module with generators , subject to the relations
Describe in terms of the structure theorem.
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Paper 4, Section II, F
2011 commentState and prove the Hilbert Basis Theorem.
Is every ring Noetherian? Justify your answer.
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Paper 1, Section I, G
2011 comment(i) State the rank-nullity theorem for a linear map between finite-dimensional vector spaces.
(ii) Show that a linear transformation of a finite-dimensional vector space is bijective if it is injective or surjective.
(iii) Let be the -vector space of all polynomials in with coefficients in . Give an example of a linear transformation which is surjective but not bijective.
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Paper 2, Section , G
2011 commentLet be an -dimensional -vector space with an inner product. Let be an -dimensional subspace of and its orthogonal complement, so that every element can be uniquely written as for and .
The reflection map with respect to is defined as the linear map
Show that is an orthogonal transformation with respect to the inner product, and find its determinant.
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Paper 4, Section I, G
2011 comment(i) Let be a vector space over a field , and subspaces of . Define the subset of , and show that and are subspaces of .
(ii) When are finite-dimensional, state a formula for in terms of and .
(iii) Let be the -vector space of all matrices over . Let be the subspace of all symmetric matrices and the subspace of all upper triangular matrices (the matrices such that whenever . Find and . Briefly justify your answer.
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Paper 1, Section II, G
2011 commentLet be finite-dimensional vector spaces over a field and a linear map.
(i) Show that is injective if and only if the image of every linearly independent subset of is linearly independent in .
(ii) Define the dual space of and the dual map .
(iii) Show that is surjective if and only if the image under of every linearly independent subset of is linearly independent in .
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Paper 2, Section II, G
2011 commentLet be a positive integer, and let be a -vector space of complex-valued functions on , generated by the set .
(i) Let for . Show that this is a positive definite Hermitian form on .
(ii) Let . Show that is a self-adjoint linear transformation of with respect to the form defined in (i).
(iii) Find an orthonormal basis of with respect to the form defined in (i), which consists of eigenvectors of .
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Paper 3, Section II, G
2011 comment(i) Let be an complex matrix and a polynomial with complex coefficients. By considering the Jordan normal form of or otherwise, show that if the eigenvalues of are then the eigenvalues of are .
(ii) Let . Write as for a polynomial with , and find the eigenvalues of
[Hint: compute the powers of .]
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Paper 4, Section II, G
2011 commentLet be an -dimensional -vector space and linear transformations. Suppose is invertible and diagonalisable, and for some real number .
(i) Show that is nilpotent, i.e. some positive power of is 0 .
(ii) Suppose that there is a non-zero vector with and . Determine the diagonal form of .
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Paper 3, Section I, H
2011 commentLet be a Markov chain with state space .
(i) What does it mean to say that has the strong Markov property? Your answer should include the definition of the term stopping time.
(ii) Show that
for a state . You may use without proof the fact that has the strong Markov property.
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Paper 4, Section I, H
2011 commentLet be a Markov chain on a state space , and let .
(i) What does the term communicating class mean in terms of this chain?
(ii) Show that .
(iii) The period of a state is defined to be
Show that if and are in the same communicating class and , then divides .
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Paper 1, Section II, H
2011 commentLet be the transition matrix for an irreducible Markov chain on the finite state space .
(i) What does it mean to say is the invariant distribution for the chain?
(ii) What does it mean to say the chain is in detailed balance with respect to ?
(iii) A symmetric random walk on a connected finite graph is the Markov chain whose state space is the set of vertices of the graph and whose transition probabilities are
where is the number of vertices adjacent to vertex . Show that the random walk is in detailed balance with respect to its invariant distribution.
(iv) Let be the invariant distribution for the transition matrix , and define an inner product for vectors by the formula
Show that the equation
holds for all vectors if and only if the chain is in detailed balance with respect to . [Here means .]
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Paper 2, Section II, H
2011 comment(i) Let be a Markov chain on the finite state space with transition matrix . Fix a subset , and let
Fix a function on such that for all , and let
where by convention. Show that
(ii) A flea lives on a polyhedron with vertices, labelled . It hops from vertex to vertex in the following manner: if one day it is on vertex , the next day it hops to one of the vertices labelled with equal probability, and it dies upon reaching vertex 1. Let be the position of the flea on day . What are the transition probabilities for the Markov chain ?
(iii) Let be the number of days the flea is alive, and let
where is a real number such that . Show that and
for . Conclude that
[Hint. Use part (i) with and a well-chosen function . ]
(iv) Show that
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Paper 2, Section I, A
2011 commentThe Legendre equation is
for and non-negative integers .
Write the Legendre equation as an eigenvalue equation for an operator in SturmLiouville form. Show that is self-adjoint and find the orthogonality relation between the eigenfunctions.
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Paper 3, Section I, A
2011 commentThe Fourier transform of the function is defined by
(i) State the inverse Fourier transform formula expressing in terms of .
(ii) State the convolution theorem for Fourier transforms.
(iii) Find the Fourier transform of the function . Hence show that the convolution of the function with itself is given by the integral expression
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Paper 4, Section I, A
2011 commentUse the method of characteristics to find a continuous solution of the equation
subject to the condition .
In which region of the plane is the solution uniquely determined?
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Paper 1, Section II, A
2011 commentLet be a real function defined on an interval with Fourier series
State and prove Parseval's theorem for and its Fourier series. Write down the formulae for and in terms of and .
Find the Fourier series of the square wave function defined on by
Hence evaluate
Using some of the above results evaluate
What is the sum of the Fourier series for at ? Comment on your answer.
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Paper 2, Section II, A
2011 commentUse a Green's function to find an integral expression for the solution of the equation
for subject to the initial conditions
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Paper 3, Section II, A
2011 commentA uniform stretched string of length , density per unit length and tension is fixed at both ends. Its transverse displacement is given by for . The motion of the string is resisted by the surrounding medium with a resistive force per unit length of .
(i) Show that the equation of motion of the string is
provided that the transverse motion can be regarded as small.
(ii) Suppose now that . Find the displacement of the string for given the initial conditions
(iii) Sketch the transverse displacement at as a function of time for .
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Paper 4, Section II, A
2011 commentLet be a two dimensional domain with boundary . Establish Green's second identity
where denotes the outward normal derivative on .
State the differential equation and boundary conditions which are satisfied by a Dirichlet Green's function for the Laplace operator on the domain , where is a fixed point in the interior of .
Suppose that on . Show that
Consider Laplace's equation in the upper half plane,
with boundary conditions where as , and as . Show that the solution is given by the integral formula
[ Hint: It might be useful to consider
for suitable . You may assume . ]
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Paper 2, Section I, 4G
2011 comment(i) Let . For , let
( is the usual Euclidean metric on .) Show that is a metric on and that the two metrics give rise to the same topology on .
(ii) Give an example of a topology on , different from the one in (i), whose induced topology (subspace topology) on the -axis is the usual topology (the one defined by the metric . Justify your answer.
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Paper 3, Section I, 3G
2011 commentLet be topological spaces, and suppose is Hausdorff.
(i) Let be two continuous maps. Show that the set
is a closed subset of .
(ii) Let be a dense subset of . Show that a continuous map is determined by its restriction to .
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Paper 1, Section II, G
2011 commentLet be a metric space with the distance function . For a subset of , its diameter is defined as .
Show that, if is compact and is an open covering of , then there exists an such that every subset with is contained in some .
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Paper 4, Section II, 13G
2011 commentLet be topological spaces and their product set. Let be the projection map.
(i) Define the product topology on . Prove that if a subset is open then is open in .
(ii) Give an example of and a closed set such that is not closed.
(iii) When is compact, show that if a subset is closed then is closed
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Paper 1, Section I, B
2011 commentOrthogonal monic polynomials are defined with respect to the inner product , where is of degree . Show that such polynomials obey a three-term recurrence relation
for appropriate choices of and .
Now suppose that is an even function of . Show that the are even or odd functions of according to whether is even or odd.
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Paper 4, Section I, B
2011 commentConsider the multistep method for numerical solution of the differential equation :
What does it mean to say that the method is of order , and that the method is convergent?
Show that the method is of order if
and give the conditions on that ensure convergence.
Hence determine for what values of and the the two-step method
is (a) convergent, and (b) of order 3 .
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Paper 1, Section II, B
2011 commentConsider a function defined on the domain . Find constants , so that for any fixed ,
is exactly satisfied for polynomials of degree less than or equal to two.
By using the Peano kernel theorem, or otherwise, show that
where . Thus show that
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Paper 2, Section II, B
2011 commentWhat is the -decomposition of a matrix A? Explain how to construct the matrices and by the Gram-Schmidt procedure, and show how the decomposition can be used to solve the matrix equation when is a square matrix.
Why is this procedure not useful for numerical decomposition of large matrices? Give a brief description of an alternative procedure using Givens rotations.
Find a -decomposition for the matrix
Is your decomposition unique? Use the decomposition you have found to solve the equation
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Paper 3, Section II, B
2011 commentA Gaussian quadrature formula provides an approximation to the integral
which is exact for all that are polynomials of degree .
Write down explicit expressions for the in terms of integrals, and explain why it is necessary that the are the zeroes of a (monic) polynomial of degree that satisfies for any polynomial of degree less than
The first such polynomials are . Show that the Gaussian quadrature formulae for are
Verify the result for by considering .
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Paper 1, Section I, H
2011 commentSuppose that and and and where and are -dimensional column vectors, and are -dimensional column vectors, and is an matrix. Here, the vector inequalities are interpreted component-wise.
(i) Show that .
(ii) Find the maximum value of
You should state any results from the course used in your solution.
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Paper 2, Section I, H
2011 commentLet be the set of nodes of a network, where 1 is the source and is the . Let denote the capacity of the arc from node to node .
(i) In the context of maximising the flow through this network, define the following terms: feasible flow, flow value, cut, cut capacity.
(ii) State and prove the max-flow min-cut theorem for network flows.
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Paper 3, Section II, H
2011 comment(i) What does it mean to say a set is convex?
(ii) What does it mean to say is an extreme point of a convex set
Let be an matrix, where . Let be an vector, and let
where the inequality is interpreted component-wise.
(iii) Show that is convex.
(iv) Let be a point in with the property that at least indices are such that . Show that is not an extreme point of . [Hint: If , then any set of vectors in is linearly dependent.]
(v) Now suppose that every set of columns of is linearly independent. Let be a point in with the property that at most indices are such that . Show that is an extreme point of .
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Paper 4, Section II, H
2011 commentA company must ship coal from four mines, labelled , to supply three factories, labelled . The per unit transport cost, the outputs of the mines, and the requirements of the factories are given below.
\begin{tabular}{c|c|c|c|c|c} & & & & & \ \hline & 12 & 3 & 5 & 2 & 34 \ \hline & 4 & 11 & 2 & 6 & 21 \ \hline & 3 & 9 & 7 & 4 & 23 \ \hline & 20 & 32 & 15 & 11 & \end{tabular}
For instance, mine can produce 32 units of coal, factory a requires 34 units of coal, and it costs 3 units of money to ship one unit of coal from to . What is the minimal cost of transporting coal from the mines to the factories?
Now suppose increased efficiency allows factory to reduce its requirement to units of coal, and as a consequence, mine reduces its output to units. By how much does the transport cost decrease?
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Paper 3, Section I, C
2011 commentA particle of mass and energy , incident from , scatters off a delta function potential at . The time independent Schrödinger equation is
where is a positive constant. Find the reflection and transmission probabilities.
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Paper 4, Section , C
2011 commentConsider the 3-dimensional oscillator with Hamiltonian
Find the ground state energy and the spacing between energy levels. Find the degeneracies of the lowest three energy levels.
[You may assume that the energy levels of the 1-dimensional harmonic oscillator with Hamiltonian
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Paper 1, Section II, C
2011 commentFor a quantum mechanical particle moving freely on a circle of length , the wavefunction satisfies the Schrödinger equation
on the interval , and also the periodicity conditions , and . Find the allowed energy levels of the particle, and their degeneracies.
The current is defined as
where is a normalized state. Write down the general normalized state of the particle when it has energy , and show that in any such state the current is independent of and . Find a state with this energy for which the current has its maximum positive value, and find a state with this energy for which the current vanishes.
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Paper 2, Section II, C
2011 commentThe quantum mechanical angular momentum operators are
Show that each of these is hermitian.
The total angular momentum operator is defined as . Show that in any state, and show that the only states where are those with no angular dependence. Verify that the eigenvalues of the operators and (whose values you may quote without proof) are consistent with these results.
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Paper 3, Section II, C
2011 commentFor an electron in a hydrogen atom, the stationary state wavefunctions are of the form , where in suitable units obeys the radial equation
Explain briefly how the terms in this equation arise.
This radial equation has bound state solutions of energy , where . Show that when , there is a solution of the form , and determine . Find the expectation value in this state.
What is the total degeneracy of the energy level with energy ?
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Paper 1, Section I,
2011 commentConsider the experiment of tossing a coin times. Assume that the tosses are independent and the coin is biased, with unknown probability of heads and of tails. A total of heads is observed.
(i) What is the maximum likelihood estimator of ?
Now suppose that a Bayesian statistician has the prior distribution for .
(ii) What is the posterior distribution for ?
(iii) Assuming the loss function is , show that the statistician's point estimate for is given by
[The distribution has density for and
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Paper 2, Section I, H
2011 commentLet be random variables with joint density function , where is an unknown parameter. The null hypothesis is to be tested against the alternative hypothesis .
(i) Define the following terms: critical region, Type I error, Type II error, size, power.
(ii) State and prove the Neyman-Pearson lemma.
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Paper 1, Section II, H
2011 commentLet be independent random variables with probability mass function , where is an unknown parameter.
(i) What does it mean to say that is a sufficient statistic for ? State, but do not prove, the factorisation criterion for sufficiency.
(ii) State and prove the Rao-Blackwell theorem.
Now consider the case where for non-negative integer and .
(iii) Find a one-dimensional sufficient statistic for .
(iv) Show that is an unbiased estimator of .
(v) Find another unbiased estimator which is a function of the sufficient statistic and that has smaller variance than . You may use the following fact without proof: has the Poisson distribution with parameter .
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Paper 3, Section II, H
2011 commentConsider the general linear model
where is a known matrix, is an unknown vector of parameters, and is an vector of independent random variables with unknown variance . Assume the matrix is invertible.
(i) Derive the least squares estimator of .
(ii) Derive the distribution of . Is an unbiased estimator of ?
(iii) Show that has the distribution with degrees of freedom, where is to be determined.
(iv) Let be an unbiased estimator of of the form for some matrix . By considering the matrix or otherwise, show that and are independent.
[You may use standard facts about the multivariate normal distribution as well as results from linear algebra, including the fact that is a projection matrix of rank , as long as they are carefully stated.]
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Paper 4, Section II, H
2011 commentConsider independent random variables with the distribution and with the distribution, where the means and variances are unknown. Derive the generalised likelihood ratio test of size of the null hypothesis against the alternative . Express the critical region in terms of the statistic and the quantiles of a beta distribution, where
[You may use the following fact: if and are independent, then
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Paper 1, Section I, D
2011 comment(i) Write down the Euler-Lagrange equations for the volume integral
where is the unit ball , and verify that the function gives a stationary value of the integral subject to the condition on the boundary.
(ii) Write down the Euler-Lagrange equations for the integral
where the dot denotes differentiation with respect to , and verify that the functions give a stationary value of the integral subject to the boundary conditions and .
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Paper 3, Section I, D
2011 commentFind, using a Lagrange multiplier, the four stationary points in of the function subject to the constraint . By considering the situation geometrically, or otherwise, identify the nature of the constrained stationary points.
How would your answers differ if, instead, the stationary points of the function were calculated subject to the constraint
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Paper 2, Section II, D
2011 comment(i) Let , where is twice differentiable and . Write down the associated Euler-Lagrange equation and show that the only solution is .
(ii) Let , where is twice differentiable and 0 . Show that only if .
(iii) Show that and deduce that the extremal value of is a global minimum.
(iv) Use the second variation of to verify that the extremal value of is a local minimum.
(v) How would your answers to part (i) differ in the case , where ? Show that the solution is not a global minimizer in this case. (You may use without proof the result .) Explain why the arguments of parts (iii) and (iv) cannot be used.
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Paper 1, Section II, H
2011 comment(i) Let be an affine variety over an algebraically closed field. Define what it means for to be irreducible, and show that if is a non-empty open subset of an irreducible , then is dense in .
(ii) Show that matrices with distinct eigenvalues form an affine variety, and are a Zariski open subvariety of affine space over an algebraically closed field.
(iii) Let be the characteristic polynomial of . Show that the matrices such that form a Zariski closed subvariety of . Hence conclude that this subvariety is all of .
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Paper 2, Section II, H
2011 comment(i) Let be an algebraically closed field, and let be an ideal in . Define what it means for to be homogeneous.
Now let be a Zariski closed subvariety invariant under ; that is, if and , then . Show that is a homogeneous ideal.
(ii) Let , and let be the graph of .
Let be the closure of in .
Write, in terms of , the homogeneous equations defining .
Assume that is an algebraically closed field of characteristic zero. Now suppose and . Find the singular points of the projective surface .
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Paper 3, Section II, H
2011 commentLet be a smooth projective curve over an algebraically closed field of characteristic 0 .
(i) Let be a divisor on .
Define , and show .
(ii) Define the space of rational differentials .
If is a point on , and a local parameter at , show that .
Use that equality to give a definition of , for . [You need not show that your definition is independent of the choice of local parameter.]
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Paper 4, Section II, H
2011 commentLet be a smooth projective curve over an algebraically closed field .
State the Riemann-Roch theorem, briefly defining all the terms that appear.
Now suppose has genus 1 , and let .
Compute for . Show that defines an isomorphism of with a smooth plane curve in which is defined by a polynomial of degree 3 .
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Paper 1, Section II, H
2011 commentAre the following statements true or false? Justify your answers.
(i) If and lie in the same path-component of , then .
(ii) If and are two points of the Klein bottle , and and are two paths from to , then and induce the same isomorphism from to .
(iii) is isomorphic to for any two spaces and .
(iv) If and are connected polyhedra and , then .
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Paper 2, Section II, H
2011 commentExplain what is meant by a covering projection. State and prove the pathlifting property for covering projections, and indicate briefly how it generalizes to a lifting property for homotopies between paths. [You may assume the Lebesgue Covering Theorem.]
Let be a simply connected space, and let be a subgroup of the group of all homeomorphisms . Suppose that, for each , there exists an open neighbourhood of such that for each other than the identity. Show that the projection is a covering projection, and deduce that .
By regarding as the set of all quaternions of modulus 1 , or otherwise, show that there is a quotient space of whose fundamental group is a non-abelian group of order
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Paper 3, Section II, H
2011 commentLet and be (finite) simplicial complexes. Explain carefully what is meant by a simplicial approximation to a continuous map . Indicate briefly how the cartesian product may be triangulated.
Two simplicial maps are said to be contiguous if, for each simplex of , there exists a simplex of such that both and are faces of . Show that:
(i) any two simplicial approximations to a given map are contiguous;
(ii) if and are contiguous, then they induce homotopic maps ;
(iii) if and are homotopic maps , then for some subdivision of there exists a sequence of simplicial maps such that is a simplicial approximation to is a simplicial approximation to and each pair is contiguous.
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Paper 4, Section II, H
2011 commentState the Mayer-Vietoris theorem, and use it to calculate, for each integer , the homology group of the space obtained from the unit disc by identifying pairs of points on its boundary whenever . [You should construct an explicit triangulation of .]
Show also how the theorem may be used to calculate the homology groups of the suspension of a connected simplicial complex in terms of the homology groups of , and of the wedge union of two connected polyhedra. Hence show that, for any finite sequence of finitely-generated abelian groups, there exists a polyhedron such that for and for . [You may assume the structure theorem which asserts that any finitely-generated abelian group is isomorphic to a finite direct sum of (finite or infinite) cyclic groups.]
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Paper 3, Section II, E
2011 commentAn electron of mass moves in a -dimensional periodic potential that satisfies the periodicity condition
where is a D-dimensional Bravais lattice. State Bloch's theorem for the energy eigenfunctions of the electron.
For a one-dimensional potential such that , give a full account of how the "nearly free electron model" leads to a band structure for the energy levels.
Explain briefly the idea of a Fermi surface and its rôle in explaining the existence of conductors and insulators.
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Paper 4, Section II, E
2011 commentA particle of charge and mass moves in a magnetic field and in an electric potential . The time-dependent Schrödinger equation for the particle's wavefunction is
where is the vector potential with . Show that this equation is invariant under the gauge transformations
where is an arbitrary function, together with a suitable transformation for which should be stated.
Assume now that , so that the particle motion is only in the and directions. Let be the constant field and let . In the gauge where show that the stationary states are given by
with
Show that is the wavefunction for a simple one-dimensional harmonic oscillator centred at position . Deduce that the stationary states lie in infinitely degenerate levels (Landau levels) labelled by the integer , with energy
A uniform electric field is applied in the -direction so that . Show that the stationary states are given by , where is a harmonic oscillator wavefunction centred now at
Show also that the eigen-energies are given by
Why does this mean that the Landau energy levels are no longer degenerate in two dimensions?
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Paper 1, Section II, E
2011 commentIn one dimension a particle of mass and momentum , is scattered by a potential where as . Incoming and outgoing plane waves of positive and negative parity are given, respectively, by
The scattering solutions to the time-independent Schrödinger equation with positive and negative parity incoming waves are and , respectively. State how the asymptotic behaviour of and can be expressed in terms of and the S-matrix denoted by
In the case where explain briefly why you expect .
The potential is given by
where is a constant. In this case, show that
where . Verify that and explain briefly the physical meaning of this result.
For , by considering the poles or zeros of show that there exists one bound state of negative parity in this potential if .
For and , show that has a pole at
where, to leading order in ,
Explain briefy the physical meaning of this result, and why you expect that .
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Paper 2, Section II, E
2011 commentA beam of particles of mass and momentum , incident along the -axis, is scattered by a spherically symmetric potential , where for large . State the boundary conditions on the wavefunction as and hence define the scattering amplitude , where is the scattering angle.
Given that, for large ,
explain how the partial-wave expansion can be used to define the phase shifts . Furthermore, given that , derive expressions for and the total cross-section in terms of the .
In a particular case is given by
where . Show that the -wave phase shift satisfies
where .
Derive an expression for the scattering length in terms of . Find the values of for which diverges and briefly explain their physical significance.
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Paper 1, Section II, J
2011 comment(i) Let be a Markov chain with finitely many states. Define a stopping time and state the strong Markov property.
(ii) Let be a Markov chain with state-space and Q-matrix
Consider the integral , the signed difference between the times spent by the chain at states and by time , and let
Derive the equation
(iii) Obtain another equation relating to .
(iv) Assuming that , where is a non-negative constant, calculate .
(v) Give an intuitive explanation why the function must have the exponential form for some .
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Paper 2, Section II, J
2011 comment(i) Explain briefly what is meant by saying that a continuous-time Markov chain is a birth-and-death process with birth rates , and death rates , .
(ii) In the case where is recurrent, find a sufficient condition on the birth and death parameters to ensure that
and express in terms of these parameters. State the reversibility property of .
Jobs arrive according to a Poisson process of rate . They are processed individually, by a single server, the processing times being independent random variables, each with the exponential distribution of rate . After processing, the job either leaves the system, with probability , or, with probability , it splits into two separate jobs which are both sent to join the queue for processing again. Let denote the number of jobs in the system at time .
(iii) In the case , evaluate , and find the expected time that the processor is busy between two successive idle periods.
(iv) What happens if ?
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Paper 3, Section II, J
2011 comment(i) Define an inhomogeneous Poisson process with rate function .
(ii) Show that the number of arrivals in an inhomogeneous Poisson process during the interval has the Poisson distribution with mean
(iii) Suppose that is a non-negative real-valued random process. Conditional on , let be an inhomogeneous Poisson process with rate function . Such a process is called a doubly-stochastic Poisson process. Show that the variance of cannot be less than its mean.
(iv) Now consider the process obtained by deleting every odd-numbered point in an ordinary Poisson process of rate . Check that
Deduce that is not a doubly-stochastic Poisson process.
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Paper 4, Section II, J
2011 commentAt an queue, the arrival times form a Poisson process of rate while service times are independent of each other and of the arrival times and have a common distribution with mean .
(i) Show that the random variables giving the number of customers left in the queue at departure times form a Markov chain.
(ii) Specify the transition probabilities of this chain as integrals in involving parameter . [No proofs are needed.]
(iii) Assuming that and the chain is positive recurrent, show that its stationary distribution has the generating function given by
for an appropriate function , to be specified.
(iv) Deduce that, in equilibrium, has the mean value
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Paper 1, Section II, A
2011 commentA function , defined for positive integer , has an asymptotic expansion for large of the following form:
What precisely does this mean?
Show that the integral
has an asymptotic expansion of the form . [The Riemann-Lebesgue lemma may be used without proof.] Evaluate the coefficients and .
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Paper 3, Section II, A
2011 commentLet
where is a complex analytic function and is a steepest descent contour from a simple saddle point of at . Establish the following leading asymptotic approximation, for large real :
Let be a positive integer, and let
where is a contour in the upper half -plane connecting to , and is real on the positive -axis with a branch cut along the negative -axis. Using the method of steepest descent, find the leading asymptotic approximation to for large .
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Paper 4, Section II, A
2011 commentDetermine the range of the integer for which the equation
has an essential singularity at .
Use the Liouville-Green method to find the leading asymptotic approximation to two independent solutions of
for large . Find the Stokes lines for these approximate solutions. For what range of is the approximate solution which decays exponentially along the positive -axis an asymptotic approximation to an exact solution with this exponential decay?
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Paper 1, Section I, C
2011 comment(i) A particle of mass and charge , at position , moves in an electromagnetic field with scalar potential and vector potential . Verify that the Lagrangian
gives the correct equations of motion.
[Note that and .]
(ii) Consider the case of a constant uniform magnetic field, with , given by , , where are Cartesian coordinates and is a constant. Find the motion of the particle, and describe it carefully.
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Paper 2, Section I, C
2011 commentThree particles, each of mass , move along a straight line. Their positions on the line containing the origin, , are and . They are subject to forces derived from the potential energy function
Obtain Lagrange's equations for the system, and show that the frequency, , of a normal mode satisfies
where . Find a complete set of normal modes for the system, and draw a diagram indicating the nature of the corresponding motions.
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Paper 3, Section I,
2011 commentThe Lagrangian for a heavy symmetric top is
State Noether's Theorem. Hence, or otherwise, find two conserved quantities linear in momenta, and a third conserved quantity quadratic in momenta.
Writing , deduce that obeys an equation of the form
where is cubic in . [You need not determine the explicit form of ]
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Paper 4, Section I, C
2011 comment(i) A dynamical system is described by the Hamiltonian . Define the Poisson bracket of two functions . Assuming the Hamiltonian equations of motion, find an expression for in terms of the Poisson bracket.
(ii) A one-dimensional system has the Hamiltonian
Show that is a constant of the motion. Deduce the form of along a classical path, in terms of the constants and .
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Paper 2, Section II, C
2011 commentDerive Euler's equations governing the torque-free and force-free motion of a rigid body with principal moments of inertia and , where . Identify two constants of the motion. Hence, or otherwise, find the equilibrium configurations such that the angular-momentum vector, as measured with respect to axes fixed in the body, remains constant. Discuss the stability of these configurations.
A spacecraft may be regarded as moving in a torque-free and force-free environment. Nevertheless, flexing of various parts of the frame can cause significant dissipation of energy. How does the angular-momentum vector ultimately align itself within the body?
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Paper 4, Section II, C
2011 commentGiven a Hamiltonian system with variables , state the definition of a canonical transformation
where and . Write down a matrix equation that is equivalent to the condition that the transformation is canonical.
Consider a harmonic oscillator of unit mass, with Hamiltonian
Write down the Hamilton-Jacobi equation for Hamilton's principal function , and deduce the Hamilton-Jacobi equation
for Hamilton's characteristic function .
Solve (1) to obtain an integral expression for , and deduce that, at energy ,
Let , and define the angular coordinate
You may assume that (2) implies
Deduce that
from which
Hence, or otherwise, show that the transformation from variables to is canonical.
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Paper 1, Section I, G
2011 commentI think of an integer with . Explain how to find using twenty questions (or less) of the form 'Is it true that ?' to which I answer yes or no.
I have watched a horse race with 15 horses. Is it possible to discover the order in which the horses finished by asking me twenty questions to which I answer yes or no?
Roughly how many questions of the yes/no type are required to discover the order in which horses finished if is large?
[You may assume that I answer honestly.]
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Paper 2, Section I, G
2011 commentI happen to know that an apparently fair coin actually has probability of heads with . I play a very long sequence of games of heads and tails in which my opponent pays me back twice my stake if the coin comes down heads and takes my stake if the coin comes down tails. I decide to bet a proportion of my fortune at the end of the th game in the st game. Determine, giving justification, the value maximizing the expected logarithm of my fortune in the long term, assuming I use the same at each game. Can it be actually disadvantageous for me to choose an (in the sense that I would be better off not playing)? Can it be actually disadvantageous for me to choose an ?
[Moral issues should be ignored.]
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Paper 3, Section I, G
2011 commentWhat is the rank of a binary linear code What is the weight enumeration polynomial of
Show that where is the rank of . Show that for all and if and only if .
Find, with reasons, the weight enumeration polynomial of the repetition code of length , and of the simple parity check code of length .
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Paper 4, Section I, G
2011 commentDescribe a scheme for sending messages based on quantum theory which is not vulnerable to eavesdropping. You may ignore engineering problems.
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Paper 1, Section II,
2011 commentDescribe the Rabin-Williams coding scheme. Show that any method for breaking it will enable us to factorise the product of two primes.
Explain how the Rabin-Williams scheme can be used for bit sharing (that is to say 'tossing coins by phone').
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Paper 2, Section II, G
2011 commentDefine a cyclic code. Show that there is a bijection between the cyclic codes of length and the factors of over the field of order 2 .
What is meant by saying that is a primitive th root of unity in a finite field extension of ? What is meant by saying that is a BCH code of length with defining set ? Show that such a code has minimum distance at least .
Suppose that is a finite field extension of in which factorises into linear factors. Show that if is a root of then is a primitive 7 th root of unity and is also a root of . Quoting any further results that you need show that the code of length 7 with defining set is the Hamming code.
[Results on the Vandermonde determinant may be used without proof provided they are quoted correctly.]
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Paper 1, Section I, E
2011 commentLight of wavelength emitted by a distant object is observed by us to have wavelength . The redshift of the object is defined by
Assuming that the object is at a fixed comoving distance from us in a homogeneous and isotropic universe with scale factor , show that
where is the time of emission and the time of observation (i.e. today).
[You may assume the non-relativistic Doppler shift formula for the shift in the wavelength of light emitted by a nearby object travelling with velocity at angle to the line of sight.]
Given that the object radiates energy per unit time, explain why the rate at which energy passes through a sphere centred on the object and intersecting the Earth is .
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Paper 2, Section I, E
2011 commentA spherically symmetric star in hydrostatic equilibrium has density and pressure , which satisfy the pressure support equation,
where is the mass within a radius . Show that this implies
Provide a justification for choosing the boundary conditions at the centre of the and at its outer radius .
Use the pressure support equation to derive the virial theorem for a star,
where is the average pressure, is the total volume of the star and is its total gravitational potential energy.
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Paper 3, Section I, E
2011 commentFor an ideal gas of fermions of mass in volume , and at temperature and chemical potential , the number density and kinetic energy are given by
where is the spin-degeneracy factor, is Planck's constant, is the single-particle energy as a function of the momentum , and
where is Boltzmann's constant.
(i) Sketch the function at zero temperature, explaining why for (the Fermi momentum). Find an expression for at zero temperature as a function of .
Assuming that a typical fermion is ultra-relativistic even at zero temperature, obtain an estimate of the energy density as a function of , and hence show that
in the ultra-relativistic limit at zero temperature.
(ii) A white dwarf star of radius has total mass , where is the proton mass and the average proton number density. On the assumption that the star's degenerate electrons are ultra-relativistic, so that applies with replaced by the average electron number density , deduce the following estimate for the star's internal kinetic energy:
By comparing this with the total gravitational potential energy, briefly discuss the consequences for white dwarf stability.
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Paper 4, Section I, 10E
2011 commentThe equilibrium number density of fermions at temperature is
where is the spin degeneracy and . For a non-relativistic gas with and , show that the number density becomes
[You may assume that for .]
Before recombination, equilibrium is maintained between neutral hydrogen, free electrons, protons and photons through the interaction
Using the non-relativistic number density , deduce Saha's equation relating the electron and hydrogen number densities,
where is the ionization energy of hydrogen. State clearly any assumptions you have made.
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Paper 1, Section II, E
2011 commentA homogeneous and isotropic universe, with scale factor , curvature parameter , energy density and pressure , satisfies the Friedmann and energy conservation equations
where , and the dot indicates a derivative with respect to cosmological time .
(i) Derive the acceleration equation
Given that the strong energy condition is satisfied, show that is a decreasing function of in an expanding universe. Show also that the density parameter satisfies
Hence explain, briefly, the flatness problem of standard big bang cosmology.
(ii) A flat homogeneous and isotropic universe is filled with a radiation fluid and a dark energy fluid , each with an equation of state of the form and density parameters today equal to and respectively. Given that each fluid independently obeys the energy conservation equation, show that the total energy density equals , where
with being the value of the Hubble parameter today. Hence solve the Friedmann equation to get
where and should be expressed in terms and . Show that this result agrees with the expected asymptotic solutions at both early and late times.
[Hint: .]
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Paper 3, Section II, E
2011 commentAn expanding universe with scale factor is filled with (pressure-free) cold dark matter (CDM) of average mass density . In the Zel'dovich approximation to gravitational clumping, the perturbed position of a CDM particle with unperturbed comoving position is given by
where is the comoving displacement.
(i) Explain why the conservation of CDM particles implies that
where is the CDM mass density. Use (1) to verify that , and hence deduce that the fractional density perturbation is, to first order,
Use this result to integrate the Poisson equation for the gravitational potential . Then use the particle equation of motion to deduce a second-order differential equation for , and hence that
[You may assume that implies and that the pressure-free acceleration equation is
(ii) A flat matter-dominated universe with background density has scale factor . The universe is filled with a pressure-free homogeneous (non-clumping) fluid of mass density , as well as cold dark matter of mass density .
Assuming that the Zel'dovich perturbation equation in this case is as in (2) but with replaced by , i.e. that
seek power-law solutions to find growing and decaying modes with
where .
Given that matter domination starts at a redshift , and given an initial perturbation , show that yields a model that is not compatible with the large-scale structure observed today.
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Paper 1, Section II, I
2011 commentLet and be manifolds and a smooth map. Define the notions critical point, critical value, regular value of . Prove that if is a regular value of , then (if non-empty) is a smooth manifold of .
[The Inverse Function Theorem may be assumed without proof if accurately stated.]
Let be the set of all real matrices and the group of all orthogonal matrices with determinant 1 . Show that is a smooth manifold and find its dimension.
Show further that is compact and that its tangent space at is given by all matrices such that .
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Paper 2, Section II, I
2011 commentLet be a smooth curve parametrized by arc-length, with for all . Define what is meant by the Frenet frame , the curvature and torsion of . State and prove the Frenet formulae.
By considering , or otherwise, show that, if for each the vectors , and are linearly dependent, then is a plane curve.
State and prove the isoperimetric inequality for regular plane curves.
[You may assume Wirtinger's inequality, provided you state it accurately.]
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Paper 3, Section II, I
2011 commentFor an oriented surface in , define the Gauss map, the second fundamental form and the normal curvature in the direction at a point .
Let be normal curvatures at in the directions , such that the angle between and is for each . Show that
where is the mean curvature of at .
What is a minimal surface? Show that if is a minimal surface, then its Gauss at each point satisfies
where depends only on . Conversely, if the identity holds at each point in , must be minimal? Justify your answer.
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Paper 4, Section II, I
2011 commentDefine what is meant by a geodesic. Let be an oriented surface. Define the geodesic curvature of a smooth curve parametrized by arc-length.
Explain without detailed proofs what are the exponential map and the geodesic polar coordinates at . Determine the derivative . Prove that the coefficients of the first fundamental form of in the geodesic polar coordinates satisfy
State the global Gauss-Bonnet formula for compact surfaces with boundary. [You should identify all terms in the formula.]
Suppose that is homeomorphic to a cylinder and has negative Gaussian curvature at each point. Prove that has at most one simple (i.e. without selfintersections) closed geodesic.
[Basic properties of geodesics may be assumed, if accurately stated.]
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Paper 1, Section I, C
2011 commentFind the fixed points of the dynamical system (with )
and determine their type as a function of .
Find the stable and unstable manifolds of the origin correct to order
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Paper 2, Section I, C
2011 commentState the Poincaré-Bendixson theorem for two-dimensional dynamical systems.
A dynamical system can be written in polar coordinates as
where and are constants with .
Show that trajectories enter the annulus .
Show that if there is a fixed point inside the annulus then and .
Use the Poincaré-Bendixson theorem to derive conditions on that guarantee the existence of a periodic orbit.
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Paper 3, Section I, C
2011 commentFor the map , with , show the following:
(i) If , then the origin is the only fixed point and is stable.
(ii) If , then the origin is unstable. There are two further fixed points which are stable for and unstable for .
(iii) If , then has the same sign as the starting value if .
(iv) If , then when . Deduce that iterates starting sufficiently close to the origin remain bounded, though they may change sign.
[Hint: For (iii) and (iv) a graphical representation may be helpful.]
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Paper 4, Section I,
2011 comment(i) Explain the use of the energy balance method for describing approximately the behaviour of nearly Hamiltonian systems.
(ii) Consider the nearly Hamiltonian dynamical system
where and are positive constants. Show that, for sufficiently small , the system has periodic orbits if , and no periodic orbits if . Show that in the first case there are two periodic orbits, and determine their approximate size and their stability.
What can you say about the existence of periodic orbits when
[You may assume that
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Paper 3, Section II, C
2011 commentExplain what is meant by a steady-state bifurcation of a fixed point of a dynamical system in , where is a real parameter.
Consider the system in , with ,
(i) Show that both the fixed point and the fixed point have a steady-state bifurcation when .
(ii) By finding the first approximation to the extended centre manifold, construct the normal form near the bifurcation point when is close to unity, and show that there is a transcritical bifurcation there. Explain why the symmetries of the equations mean that the bifurcation at must be of pitchfork type.
(iii) Show that two fixed points with exist in the range . Show that the solution with is stable. Identify the bifurcation that occurs at .
(iv) Draw a sketch of the values of at the fixed points as functions of , indicating the bifurcation points and the regions where each branch is stable. [Detailed calculations are not required.]
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Paper 4, Section II, C
2011 comment(i) State and prove Lyapunov's First Theorem, and state (without proof) La Salle's Invariance Principle. Show by example how the latter result can be used to prove asymptotic stability of a fixed point even when a strict Lyapunov function does not exist.
(ii) Consider the system
Show that the origin is asymptotically stable and that the basin of attraction of the origin includes the region .
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Paper 1, Section II, C
2011 commentIn the Landau-Ginzburg model of superconductivity, the energy of the system is given, for constants and , by
where is the time-independent magnetic field derived from the vector potential , and is the wavefunction of the charge carriers, which have mass and charge .
Describe the physical meaning of each of the terms in the integral.
Explain why in a superconductor one must choose and . Find an expression for the number density of the charge carriers in terms of and .
Show that the energy is invariant under the gauge transformations
Assuming that the number density is uniform, show that, if is a minimum under variations of , then
where .
Find a formula for and use it to explain why there cannot be a magnetic field inside the bulk of a superconductor.
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Paper 3, Section II, C
2011 commentExplain how time-dependent distributions of electric charge and current can be combined into a four-vector that obeys .
This current generates a four-vector potential . Explain how to find in the gauge .
A small circular loop of wire of radius is centred at the origin. The unit vector normal to the plane of the loop is . A current flows in the loop. Find the three-vector potential to leading order in .
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Paper 4, Section II, C
2011 commentSuppose that there is a distribution of electric charge given by the charge density . Develop the multipole expansion, up to quadrupole terms, for the electrostatic potential and define the dipole and quadrupole moments of the charge distribution.
A tetrahedron has a vertex at where there is a point charge of strength . At each of the other vertices located at and there is a point charge of strength .
What is the dipole moment of this charge distribution?
What is the quadrupole moment?
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Paper 1, Section II, B
2011 commentThe steady two-dimensional boundary-layer equations for flow primarily in the direction are
A thin, steady, two-dimensional jet emerges from a point at the origin and flows along the -axis in a fluid at rest far from the -axis. Show that the momentum flux
is independent of position along the jet. Deduce that the thickness of the jet increases along the jet as , while the centre-line velocity decreases as .
A similarity solution for the jet is sought with a streamfunction of the form
Derive the nonlinear third-order non-dimensional differential equation governing , and write down the boundary and normalisation conditions which must be applied.
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Paper 2, Section II, B
2011 commentThe energy equation for the motion of a viscous, incompressible fluid states that
Interpret each term in this equation and explain the meaning of the symbols used.
Consider steady rectilinear flow in a (not necessarily circular) pipe having rigid stationary walls. Deduce a relation between the viscous dissipation per unit length of the pipe, the pressure gradient , and the volume flux .
Starting from the Navier-Stokes equations, calculate the velocity field for steady rectilinear flow in a circular pipe of radius . Using the relationship derived above, or otherwise, find the viscous dissipation per unit length of this flow in terms of .
[Hint: In cylindrical polar coordinates,
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Paper 3, Section II, B
2011 commentIf is harmonic, i.e. if , show that
satisfies the incompressibility condition and the Stokes equation. Show that the stress tensor is
Consider the Stokes flow corresponding to
where are the components of a constant vector . Show that on the sphere the normal component of velocity vanishes and the surface traction is in the normal direction. Hence deduce that the drag force on the sphere is given by
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Paper 4, Section II, B
2011 commentA viscous fluid flows along a slowly varying thin channel between no-slip surfaces at and under the action of a pressure gradient . After explaining the approximations and assumptions of lubrication theory, including a comment on the reduced Reynolds number, derive the expression for the volume flux
as well as the equation
In peristaltic pumping, the surface has a periodic form in space which propagates at a constant speed , i.e. , and no net pressure gradient is applied, i.e. the pressure gradient averaged over a period vanishes. Show that the average flux along the channel is given by
where denotes an average over one period.
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Paper 1, Section I, E
2011 commentShow that the following integral is well defined:
Express in terms of a combination of hypergeometric functions.
[You may assume without proof that the hypergeometric function can be expressed in the form
for appropriate restrictions on . Furthermore,
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Paper 2, Section I, E
2011 commentFind the two complex-valued functions and such that all of the following hold:
(i) and are analytic for and respectively, where .
(ii) .
(iii) .
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Paper 3, Section I, E
2011 commentExplain the meaning of in the Weierstrass canonical product formula
Show that
Deduce that
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Paper 4, Section , E
2011 commentLet be defined by
Let be defined by
where denotes principal value integral and the contour is the negative imaginary axis.
By computing , obtain a formula for the analytic continuation of for .
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Paper 1, Section II, E
2011 comment(i) By assuming the validity of the Fourier transform pair, prove the validity of the following transform pair:
where is an arbitrary complex constant and is the union of the two rays arg and with the orientation shown in the figure below:

The contour .
(ii) Verify that the partial differential equation
can be rewritten in the following form:
Consider equation (2) supplemented with the conditions
By using equations (1a) and (3), show that
where
Part II, List of Questions
[TURN OVER
Use (1b) to invert equation (5) and furthermore show that
Hence determine the constant so that the solution of equation (2), with the conditions (4) and with the condition that either or is given, can be expressed in terms of an integral involving and either or .
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Paper 2, Section II, E
2011 commentConsider the following sum related to Riemann's zeta function:
where denotes the integer part of .
(i) By using an appropriate branch cut, show that
where is the circle in the complex -plane centred at with radius , .
(ii) Use the above representation to show that, for and ,
where is defined in (i) and the curves are the following semi-circles in the right half complex -plane:

The curves and .
Part II, 2011 List of Questions
[TURN OVER
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Paper 1, Section II, 18H
2011 commentLet be a field.
(i) Let and be two finite extensions of . When the degrees of these two extensions are equal, show that every -homomorphism is an isomorphism. Give an example, with justification, of two finite extensions and of , which have the same degrees but are not isomorphic over .
(ii) Let be a finite extension of . Let and be two finite extensions of . Show that if and are isomorphic as extensions of then they are isomorphic as extensions of . Prove or disprove the converse.
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Paper 2, Section II, H
2011 commentLet be the function field in two variables . Let , and be the subfield of of all rational functions in and
(i) Let , which is a subfield of . Show that is a quadratic extension.
(ii) Show that is cyclic of order , and is Galois. Determine the Galois .
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Paper 3, Section II, H
2011 commentLet and be the cyclotomic field generated by the th roots of unity. Let with , and consider .
(i) State, without proof, the theorem which determines .
(ii) Show that is a Galois extension and that is soluble. [When using facts about general Galois extensions and their generators, you should state them clearly.]
(iii) When is prime, list all possible degrees , with justification.
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Paper 4, Section II, H
2011 commentLet be a field of characteristic 0 , and let be an irreducible quartic polynomial over . Let be its roots in an algebraic closure of , and consider the Galois group (the group for a splitting field of over ) as a subgroup of (the group of permutations of .
Suppose that contains .
(i) List all possible up to isomorphism. [Hint: there are 4 cases, with orders 4 , 8,12 and 24.]
(ii) Let be the resolvent cubic of , i.e. a cubic in whose roots are and . Construct a natural surjection , and find in each of the four cases found in (i).
(iii) Let be the discriminant of . Give a criterion to determine in terms of and the factorisation of in .
(iv) Give a specific example of where is abelian.
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Paper 1, Section II, 37D
2011 commentConsider a metric of the form
Let describe an affinely-parametrised geodesic, where . Write down explicitly the Lagrangian
with , using the given metric. Hence derive the four geodesic equations. In particular, show that
By comparing these equations with the standard form of the geodesic equation, show that and derive the other Christoffel symbols.
The Ricci tensor, , is defined by
By considering the case , show that the vacuum Einstein field equations imply
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Paper 2, Section II, 36D
2011 commentThe curvature tensor satisfies
for any covariant vector field . Hence express in terms of the Christoffel symbols and their derivatives. Show that
Further, by setting , deduce that
Using local inertial coordinates or otherwise, obtain the Bianchi identities.
Define the Ricci tensor in terms of the curvature tensor and show that it is symmetric. [You may assume that .] Write down the contracted Bianchi identities.
In certain spacetimes of dimension takes the form
Obtain the Ricci tensor and curvature scalar. Deduce, under some restriction on which should be stated, that is a constant.
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Paper 4, Section II, D
2011 commentThe metric of the Schwarzschild solution is
Show that, for an incoming radial light ray, the quantity
is constant.
Express in terms of and . Determine the light-cone structure in these coordinates, and use this to discuss the nature of the apparent singularity at .
An observer is falling radially inwards in the region . Assuming that the metric for is again given by , obtain a bound for , where is the proper time of the observer, in terms of . Hence, or otherwise, determine the maximum proper time that can elapse between the events at which the observer crosses and is torn apart at .
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Paper 1, Section I, G
2011 commentLet be a finite subgroup of and let be the set of unit vectors that are fixed by some non-identity element of . Show that the group permutes the unit vectors in and that has at most three orbits. Describe these orbits when is the group of orientation-preserving symmetries of a regular dodecahedron.
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Paper 2, Section ,
2011 commentLet and be two rotations of the Euclidean plane about centres and respectively. Show that the conjugate is also a rotation and find its fixed point. When do and commute? Show that the commutator is a translation.
Deduce that any group of orientation-preserving isometries of the Euclidean plane either fixes a point or is infinite.
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Paper 3, Section I,
2011 commentDefine a Kleinian group.
Give an example of a Kleinian group that is a free group on two generators and explain why it has this property.
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Paper 4, Section I, G
2011 commentDefine inversion in a circle on the Riemann sphere. You should show from your definition that inversion in exists and is unique.
Prove that the composition of an even number of inversions is a Möbius transformation of the Riemann sphere and that every Möbius transformation is the composition of an even number of inversions.
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Paper 1, Section II, G
2011 commentProve that a group of Möbius transformations is discrete if, and only if, it acts discontinuously on hyperbolic 3 -space.
Let be the set of Möbius transformations with
Show that is a group and that it acts discontinuously on hyperbolic 3-space. Show that contains transformations that are elliptic, parabolic, hyperbolic and loxodromic.
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Paper 4, Section II, G
2011 commentDefine a lattice in and the rank of such a lattice.
Let be a rank 2 lattice in . Choose a vector with as small as possible. Then choose with as small as possible. Show that .
Suppose that is the unit vector . Draw the region of possible values for . Suppose that also equals . Prove that
for some integers with .
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Paper 1, Section II, F
2011 commentLet be a bipartite graph with vertex classes and . What is a matching from to ?
Show that if for all then contains a matching from to .
Let be a positive integer. Show that if for all then contains a set of independent edges.
Show that if 0 is not an eigenvalue of then contains a matching from to .
Suppose now that and that does contain a matching from to . Must it be the case that 0 is not an eigenvalue of ? Justify your answer.
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Paper 2, Section II, F
2011 commentWhat does it mean to say that a graph is -colourable? Define the chromatic number of a graph , and the chromatic number of a closed surface .
State the Euler-Poincaré formula relating the numbers of vertices, edges and faces in a drawing of a graph on a closed surface of Euler characteristic . Show that if then
Find, with justification, the chromatic number of the Klein bottle . Show that if is a triangle-free graph which can be drawn on the Klein bottle then .
[You may assume that the Klein bottle has Euler characteristic 0 , and that can be drawn on the Klein bottle but cannot. You may use Brooks's theorem.]
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Paper 3, Section II,
2011 commentDefine the Turán graph . State and prove Turán's theorem. Hence, or otherwise, find .
Let be a bipartite graph with vertices in each class. Let be an integer, , and assume . Show that contains a set of independent edges.
[Hint: Suppose contains a set of a independent edges but no set of a independent edges. Let be the set of vertices of the edges in and let be the set of edges in with precisely one vertex in ; consider
Hence, or otherwise, show that if is a triangle-free tripartite graph with vertices in each class then .
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Paper 4, Section II, F
2011 comment(i) Given a positive integer , show that there exists a positive integer such that, whenever the edges of the complete graph are coloured with colours, there exists a monochromatic triangle.
Denote the least such by . Show that for all .
(ii) You may now assume that and .
Let denote the graph of order 4 consisting of a triangle together with one extra edge. Given a positive integer , let denote the least positive integer such that, whenever the edges of the complete graph are coloured with colours, there exists a monochromatic copy of . By considering the edges from one vertex of a monochromatic triangle in , or otherwise, show that . By exhibiting a blue-yellow colouring of the edges of with no monochromatic copy of , show that in fact .
What is Justify your answer.
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Paper 1, Section II, A
2011 commentDefine a finite-dimensional integrable system and state the Arnold-Liouville theorem.
Consider a four-dimensional phase space with coordinates , where and is periodic with period . Let the Hamiltonian be
Show that the corresponding Hamilton equations form an integrable system.
Determine the sign of the constant so that the motion is periodic on the surface . Demonstrate that in this case, the action variables are given by
where are positive constants which you should determine.
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Paper 2, Section II, A
2011 commentConsider the Poisson structure
where are polynomial functionals of . Assume that tend to zero as .
(i) Show that .
(ii) Write down Hamilton's equations for corresponding to the following Hamiltonians:
(iii) Calculate the Poisson bracket , and hence or otherwise deduce that the following overdetermined system of partial differential equations for is compatible:
[You may assume that the Jacobi identity holds for (1).]
(iv) Find a symmetry of (3) generated by for some constant which should be determined. Construct a vector field corresponding to the one parameter group
where should be determined from the symmetry requirement. Find the Lie algebra generated by the vector fields .
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Paper 3, Section II, A
2011 commentLet and be matrix-valued functions. Consider the following system of overdetermined linear partial differential equations:
where is a column vector whose components depend on . Using the consistency condition of this system, derive the associated zero curvature representation (ZCR)
where denotes the usual matrix commutator.
(i) Let
Find a partial differential equation for which is equivalent to the .
(ii) Assuming that and in do not depend on , show that the trace of does not depend on , where is any positive integer. Use this fact to construct a first integral of the ordinary differential equation
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Paper 1, Section II, G
2011 commentState a version of the Stone-Weierstrass Theorem for real-valued functions on a compact metric space.
Suppose that is a continuous function. Show that may be uniformly approximated by functions of the form with continuous.
Let be Banach spaces and suppose that is a bounded linear operator. What does it mean to say that is finite-rank? What does it mean to say that is compact? Give an example of a bounded linear operator from to itself which is not compact.
Suppose that is a sequence of finite-rank operators and that in the operator norm. Briefly explain why the are compact. Show that is compact.
Hence, show that the integral operator defined by
is compact.
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Paper 2, Section II, G
2011 commentState and prove the Baire Category Theorem. Let be a function. For , define
Show that is continuous at if and only if .
Show that for any the set is open.
Hence show that the set of points at which is continuous cannot be precisely the set of rationals.
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Paper 3, Section II, G
2011 commentLet be a complex Hilbert space with orthonormal basis Let be a bounded linear operator. What is meant by the spectrum of ?
Define by setting for . Show that has a unique extension to a bounded, self-adjoint linear operator on . Determine the norm . Exhibit, with proof, an element of .
Show that has no eigenvectors. Is compact?
[General results from spectral theory may be used without proof. You may also use the fact that if a sequence satisfies a linear recurrence with , , then it has the form or , where and .]
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Paper 4, Section II, G
2011 commentState Urysohn's Lemma. State and prove the Tietze Extension Theorem.
Let be two topological spaces. We say that the extension property holds if, whenever is a closed subset and is a continuous map, there is a continuous function with .
For each of the following three statements, say whether it is true or false. Briefly justify your answers.
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If is a metric space and then the extension property holds.
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If is a compact Hausdorff space and then the extension property holds.
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If is an arbitrary topological space and then the extension property holds.
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Paper 1, Section II, H
2011 commentGive the inductive and synthetic definitions of ordinal addition, and prove that they are equivalent.
Which of the following assertions about ordinals and are always true, and which can be false? Give proofs or counterexamples as appropriate.
(i) .
(ii) .
(iii) If then .
(iv) If then .
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Paper 2, Section II, H
2011 commentState and prove Zorn's Lemma. [You may assume Hartogs' Lemma.] Where in your argument have you made use of the Axiom of Choice?
Show that every real vector space has a basis.
Let be a real vector space having a basis of cardinality . What is the cardinality of ? Justify your answer.
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Paper 3, Section II, H
2011 commentState and prove the Upward Löwenheim-Skolem Theorem.
[You may assume the Compactness Theorem, provided that you state it clearly.]
A total ordering is called dense if for any there exists with . Show that a dense total ordering (on more than one point) cannot be a well-ordering.
For each of the following theories, either give axioms, in the language of posets, for the theory or prove carefully that the theory is not axiomatisable in the language of posets.
(i) The theory of dense total orderings.
(ii) The theory of countable dense total orderings.
(iii) The theory of uncountable dense total orderings.
(iv) The theory of well-orderings.
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Paper 4, Section II, H
2011 commentDefine the sets for ordinals . Show that each is transitive. Show also that whenever . Prove that every set is a member of some .
For which ordinals does there exist a set such that the power-set of has rank ? [You may assume standard properties of rank.]
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Paper 1, Section I, B
2011 commentA proposed model of insect dispersal is given by the equation
where is the density of insects and and are constants.
Interpret the term on the right-hand side.
Explain why a solution of the form
where is a positive constant, can potentially represent the dispersal of a fixed number of insects initially localised at the origin.
Show that the equation (1) can be satisfied by a solution of the form (2) if and find the corresponding function .
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Paper 2, Section I, B
2011 commentA population with variable growth and harvesting is modelled by the equation
where and are positive constants.
Given that , show that a non-zero steady state exists if , where is to be determined.
Show using a cobweb diagram that, if , a non-zero steady state may be attained only if the initial population satisfies , where should be determined explicitly and should be specified as a root of an algebraic equation.
With reference to the cobweb diagram, give an additional criterion that implies that is a sufficient condition, as well as a necessary condition, for convergence to a non-zero steady state.
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Paper 3, Section I, B
2011 commentThe dynamics of a directly transmitted microparasite can be modelled by the system
where and are positive constants and and are respectively the numbers of susceptible, infected and immune (i.e. infected by the parasite, but showing no further symptoms of infection) individuals in a population of size , independent of , where .
Consider the possible steady states of these equations. Show that there is a threshold population size such that if there is no steady state with the parasite maintained in the population. Show that in this case the number of infected and immune individuals decreases to zero for all possible initial conditions.
Show that for there is a possible steady state with and , and find expressions for and .
By linearising the equations for and about the steady state and , derive a quadratic equation for the possible growth or decay rate in terms of and and hence show that the steady state is stable.
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Paper 4, Section I, B
2011 commentA neglected flower garden contains marigolds in the summer of year . On average each marigold produces seeds through the summer. Seeds may germinate after one or two winters. After three winters or more they will not germinate. Each winter a fraction of all seeds in the garden are eaten by birds (with no preference to the age of the seed). In spring a fraction of seeds that have survived one winter and a fraction of seeds that have survived two winters germinate. Finite resources of water mean that the number of marigolds growing to maturity from germinating seeds is , where is an increasing function such that is a decreasing function of and as
Show that satisfies the equation
Write down an equation for the number of marigolds in a steady state. Show graphically that there are two solutions, one with and the other with if
Show that the steady-state solution is unstable to small perturbations in this case.
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Paper 2, Section II, B
2011 commentConsider a population subject to the following birth-death process. When the number of individuals in the population is , the probability of an increase from to in unit time is and the probability of a decrease from to is , where and are constants.
Show that the master equation for , the probability that at time the population has members, is
Show that , the mean number of individuals in the population, satisfies
Deduce that, in a steady state,
where is the standard deviation of . When is the minus sign admissable?
Show how a Fokker-Planck equation of the form
may be derived under conditions to be explained, where the functions and should be evaluated.
In the case and , find the leading-order approximation to such that . Defining the new variable , where , approximate by and by . Solve for in the steady-state limit and deduce leading-order estimates for and .
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Paper 3, Section II, B
2011 commentThe number density of a population of amoebae is . The amoebae exhibit chemotaxis and are attracted to high concentrations of a chemical which has concentration . The equations governing and are
where the constants and are all positive.
(i) Give a biological interpretation of each term in these equations and discuss the sign of .
(ii) Show that there is a non-trivial (i.e. ) steady-state solution for and , independent of , and show further that it is stable to small disturbances that are also independent of .
(iii) Consider small spatially varying disturbances to the steady state, with spatial structure such that , where is any disturbance quantity. Show that if such disturbances also satisfy , where is a constant, then satisfies a quadratic equation, to be derived. By considering the conditions required for to be a possible solution of this quadratic equation, or otherwise, deduce that instability is possible if
where .
(iv) Explain briefly how your conclusions might change if an additional geometric constraint implied that , where is a given constant.
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Paper 1, Section II, F
2011 commentCalculate the class group for the field .
[You may use any general theorem, provided that you state it accurately.]
Find all solutions in of the equation .
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Paper 2, Section II, F
2011 comment(i) Suppose that is a square-free integer. Describe, with justification, the ring of integers in the field .
(ii) Show that and that is not the ring of integers in this field.
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Paper 4, Section II, F
2011 comment(i) Prove that the ring of integers in a real quadratic field contains a non-trivial unit. Any general results about lattices and convex bodies may be assumed.
(ii) State the general version of Dirichlet's unit theorem.
(iii) Show that for is a fundamental unit in .
[You may not use results about continued fractions unless you prove them.]
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Paper 1, Section I, I
2011 commentProve that, under the action of , every positive definite binary quadratic form of discriminant , with integer coefficients, is equivalent to
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Paper 4, Section , I
2011 comment(i) Prove that there are infinitely many primes.
(ii) Prove that arbitrarily large gaps can occur between consecutive primes.
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Paper 2, Section I, I
2011 comment(i) Find a primitive root modulo
(ii) Let be a prime of the form for some integer . Prove that every quadratic non-residue modulo is a primitive root modulo .
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Paper 3, Section I, I
2011 comment(i) State Lagrange's Theorem, and prove that, if is an odd prime,
(ii) Still assuming is an odd prime, prove that
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Paper 3, Section II, I
2011 commentLet be the Riemann zeta function, and put with .
(i) If , prove that
where the product is taken over all primes .
(ii) Assuming that, for , we have
prove that has an analytic continuation to the half plane .
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Paper 4, Section II, I
2011 comment(i) Prove the law of reciprocity for the Jacobi symbol. You may assume the law of reciprocity for the Legendre symbol.
(ii) Let be an odd positive integer which is not a square. Prove that there exists an odd prime with .
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Paper 1, Section II, A
2011 commentThe nine-point method for the Poisson equation (with zero Dirichlet boundary conditions) in a square, reads
where , for all .
(i) By arranging the two-dimensional arrays and into column vectors and respectively, the linear system above takes the matrix form . Prove that, regardless of the ordering of the points on the grid, the matrix is symmetric and negative definite.
(ii) Formulate the Jacobi method with relaxation for solving the above linear system.
(iii) Prove that the iteration converges if the relaxation parameter is equal to
[You may quote without proof any relevant result about convergence of iterative methods.]
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Paper 2, Section II, A
2011 commentLet be a real matrix with linearly independent eigenvectors. The eigenvalues of can be calculated from the sequence , which is generated by the power method
where is a real nonzero vector.
(i) Describe the asymptotic properties of the sequence in the case that the eigenvalues of satisfy , and the eigenvectors are of unit length.
(ii) Present the implementation details for the power method for the setting in (i) and define the Rayleigh quotient.
(iii) Let be the matrix
where is real and nonzero. Find an explicit expression for
Let the sequence be generated by the power method as above. Deduce from your expression for that the first and second components of tend to zero as . Further show that this implies as .
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Paper 3, Section II, A
2011 comment(i) The difference equation
where , is the basic equation used in the second-order AdamsBashforth method and can be employed to approximate a solution of the diffusion equation . Prove that, as with constant , the local error of the method is .
(ii) By applying the Fourier stability test, show that the above method is stable if and only if .
(iii) Define the leapfrog scheme to approximate the diffusion equation and prove that it is unstable for every choice of .
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Paper 4, Section II, A
2011 comment(i) Consider the Poisson equation
with the periodic boundary conditions
and the normalization condition
Moreover, is analytic and obeys the periodic boundary conditions
Derive an explicit expression of the approximation of a solution by means of a spectral method. Explain the term convergence with spectral speed and state its validity for the approximation of .
(ii) Consider the second-order linear elliptic partial differential equation
with the periodic boundary conditions and normalization condition specified in (i). Moreover, and are given by
[Note that is a positive analytic periodic function.]
Construct explicitly the linear algebraic system that arises from the implementation of a spectral method to the above equation.
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Paper 2, Section II, K
2011 commentConsider an optimal stopping problem in which the optimality equation takes the form
, and where for all . Let denote the stopping set of the onestep-look-ahead rule. Show that if is closed (in a sense you should explain) then the one-step-look-ahead rule is optimal.
biased coins are to be tossed successively. The probability that the th coin toss will show a head is known to be . At most once, after observing a head, and before tossing the next coin, you may guess that you have just seen the last head (i.e. that all subsequent tosses will show tails). If your guess turns out to be correct then you win .
Suppose that you have not yet guessed 'last head', and the th toss is a head. Show that it cannot be optimal to guess that this is the last head if
where .
Suppose that . Show that it is optimal to guess that the last head is the first head (if any) to occur after having tossed at least coins, where when is large.
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Paper 3, Section II, 28K
2011 commentAn observable scalar state variable evolves as Let controls be determined by a policy and define
Show that it is possible to express in terms of , which satisfies the recurrence
with .
Deduce that is defined as
By considering the policy which takes , show that .
Give an alternative description of in closed-loop form.
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Paper 4, Section II, K
2011 commentDescribe the type of optimal control problem that is amenable to analysis using Pontryagin's Maximum Principle.
A firm has the right to extract oil from a well over the interval . The oil can be sold at price per unit. To extract oil at rate when the remaining quantity of oil in the well is incurs cost at rate . Thus the problem is one of maximizing
subject to . Formulate the Hamiltonian for this problem.
Explain why , the adjoint variable, has a boundary condition .
Use Pontryagin's Maximum Principle to show that under optimal control
and
Find the oil remaining in the well at time , as a function of , and ,
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Paper 1, Section II, A
2011 commentLet , be a smooth real-valued function which maps into . Consider the initial value problem for the equation
for the unknown function .
(i) Use the method of characteristics to solve the initial value problem, locally in time.
(ii) Let on . Use the method of characteristics to prove that remains non-negative (as long as it exists).
(iii) Let be smooth. Prove that
as long as the solution exists.
(iv) Let be independent of , namely , where is smooth and realvalued. Give the explicit solution of the initial value problem.
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Paper 2, Section II, A
2011 commentConsider the Schrödinger equation
where is a smooth real-valued function.
Prove that, for smooth solutions, the following equations are valid for all :
(i)
(ii)
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Paper 3, Section II, A
2011 comment(a) State the local existence theorem of a classical solution of the Cauchy problem
where is a smooth curve in .
(b) Solve, by using the method of characteristics,
where is a constant. What is the maximal domain of existence in which is a solution of the Cauchy problem?
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Paper 4, Section II, A
2011 commentConsider the functional
where is a bounded domain in with smooth boundary and is smooth. Assume that is convex in for all and that there is a such that
(i) Prove that is well-defined on , bounded from below and strictly convex. Assume without proof that is weakly lower-semicontinuous. State this property. Conclude the existence of a unique minimizer of .
(ii) Which elliptic boundary value problem does the minimizer solve?
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Paper 1, Section II, D
2011 commentTwo individual angular momentum states , acted on by and respectively, can be combined to form a combined state . What is the combined angular momentum operator in terms of and ? [Units in which are to be used throughout.]
Defining raising and lowering operators , where , find an expression for in terms of and . Show that this implies
Write down the state with and with eigenvalue in terms of the individual angular momentum states. From this starting point, calculate the combined state with eigenvalues and in terms of the individual angular momentum states.
If and and the combined system is in the state , what is the probability of measuring the eigenvalues of individual angular momentum states to be and 0 , respectively?
[You may assume without proof that standard angular momentum states are joint eigenstates of and , obeying
and that
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Paper 2, Section II, D
2011 commentA quantum system has energy eigenstates with eigenvalues . An observable is such that .
(a) What is the commutator of with the Hamiltonian ?
(b) Given , consider the state
Determine:
(i) The probability of measuring to be .
(ii) The probability of measuring energy followed by another immediate measurement of energy .
(iii) The average of many separate measurements of , each measurement being on a state , as .
(c) Given and for , consider the state
where .
(i) Show that the probability of measuring an eigenvalue of is
where and are integers that you should find.
(ii) Show that is , where and are integers that you should find.
(iii) Given that is measured to be at time , write down the state after a time has passed. What is then the subsequent probability at time of measuring the energy to be ?
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Paper 3, Section II, D
2011 commentThe Pauli matrices , with
are used to represent angular momentum operators with respect to basis states and corresponding to spin up and spin down along the -axis. They satisfy
(i) How are and represented? How is the spin operator s related to and ? Check that the commutation relations between the spin operators are as desired. Check that acting on a spin one-half state has the correct eigenvalue.
What are the states obtained by applying to the eigenstates and of ?
(ii) Let be the space of states for a spin one-half system. Consider a combination of three such systems with states belonging to and spin operators acting on each subsystem denoted by with . Find the eigenvalues of the operators
of the state
(iii) Consider now whether these outcomes for measurements of particular combinations of the operators in the state could be reproduced by replacing the spin operators with classical variables which take values according to some probabilities. Assume that these variables are identical to the quantum measurements of on . Show that classically this implies a unique possibility for
and find its value.
State briefly how this result could be used to experimentally test quantum mechanics.
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Paper 4, Section II, D
2011 commentThe quantum-mechanical observable has just two orthonormal eigenstates and with eigenvalues and 1 , respectively. The operator is defined by , where
Defining orthonormal eigenstates of to be and with eigenvalues , , respectively, consider a perturbation to first order in for the states
where are complex coefficients. The real eigenvalues are also expanded to first order in :
From first principles, find .
Working exactly to all orders, find the real eigenvalues directly. Show that the exact eigenvectors of may be taken to be of the form
finding and the real numerical coefficient in the process.
By expanding the exact expressions, again find , verifying the perturbation theory results above.
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Paper 1, Section II, K
2011 commentDefine admissible, Bayes, minimax decision rules.
A random vector has independent components, where has the normal distribution when the parameter vector takes the value . It is required to estimate by a point , with loss function . What is the risk function of the maximum-likelihood estimator Show that is dominated by the estimator .
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Paper 2, Section II, K
2011 commentRandom variables are independent and identically distributed from the normal distribution with unknown mean and unknown precision (inverse variance) . Show that the likelihood function, for data , is
where and .
A bivariate prior distribution for is specified, in terms of hyperparameters , as follows. The marginal distribution of is , with density
and the conditional distribution of , given , is normal with mean and precision .
Show that the conditional prior distribution of , given , is
Show that the posterior joint distribution of , given , has the same form as the prior, with updated hyperparameters which you should express in terms of the prior hyperparameters and the data.
[You may use the identity
where and .]
Explain how you could implement Gibbs sampling to generate a random sample from the posterior joint distribution.
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Paper 3, Section II,
2011 commentRandom variables are independent and identically distributed from the exponential distribution , with density function
when the parameter takes value . The following experiment is performed. First is observed. Thereafter, if have been observed , a coin having probability of landing heads is tossed, where is a known function and the coin toss is independent of the 's and previous tosses. If it lands heads, no further observations are made; if tails, is observed.
Let be the total number of 's observed, and . Write down the likelihood function for based on data , and identify a minimal sufficient statistic. What does the likelihood principle have to say about inference from this experiment?
Now consider the experiment that only records . Show that the density function of has the form
Assuming the function is twice differentiable and that both and vanish at 0 and , show that is an unbiased estimator of , and find its variance.
Stating clearly any general results you use, deduce that
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Paper 4, Section II, K
2011 commentWhat does it mean to say that a random vector has a multivariate normal distribution?
Suppose has the bivariate normal distribution with mean vector , and dispersion matrix
Show that, with is independent of , and thus that the conditional distribution of given is normal with mean and variance .
For are independent and identically distributed with the above distribution, where all elements of and are unknown. Let
where .
The sample correlation coefficient is . Show that the distribution of depends only on the population correlation coefficient .
Student's -statistic (on degrees of freedom) for testing the null hypothesis is
where and . Its density when is true is
where is a constant that need not be specified.
Express in terms of , and hence derive the density of when .
How could you use the sample correlation to test the hypothesis ?
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Paper 1, Section II,
2011 comment(i) Let be a measure space and let . For a measurable function , let . Give the definition of the space . Prove that forms a Banach space.
[You may assume that is a normed vector space. You may also use in your proof any other result from the course provided that it is clearly stated.]
(ii) Show that convergence in probability implies convergence in distribution.
[Hint: Show the pointwise convergence of the characteristic function, using without proof the inequality for .]
(iii) Let be a given real-valued sequence such that . Let be a sequence of independent standard Gaussian random variables defined on some probability space . Let
Prove that there exists a random variable such that in .
(iv) Specify the distribution of the random variable defined in part (iii), justifying carefully your answer.
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Paper 2, Section II,
2011 comment(i) Define the notions of a -system and a -system. State and prove Dynkin's lemma.
(ii) Let and denote two finite measure spaces. Define the algebra and the product measure . [You do not need to verify that such a measure exists.] State (without proof) Fubini's Theorem.
(iii) Let be a measure space, and let be a non-negative Borel-measurable function. Let be the subset of defined by
Show that , where denotes the Borel -algebra on . Show further that
where is Lebesgue measure.
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Paper 3, Section II, 25K
2011 comment(i) State and prove Kolmogorov's zero-one law.
(ii) Let be a finite measure space and suppose that is a sequence of events such that for all . Show carefully that , where .
(iii) Let be a sequence of independent and identically distributed random variables such that and . Let and consider the event defined by
Prove that there exists such that for all large enough, . Any result used in the proof must be stated clearly.
(iv) Prove using the results above that occurs infinitely often, almost surely. Deduce that
almost surely.
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Paper 4, Section II, K
2011 comment(i) State and prove Fatou's lemma. State and prove Lebesgue's dominated convergence theorem. [You may assume the monotone convergence theorem.]
In the rest of the question, let be a sequence of integrable functions on some measure space , and assume that almost everywhere, where is a given integrable function. We also assume that as .
(ii) Show that and that , where and denote the positive and negative parts of a function .
(iii) Here we assume also that . Deduce that .
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Paper 1, Section II, I
2011 commentLet be a finite group and its centre. Suppose that has order and has order . Suppose that is a complex irreducible representation of degree
(i) For , show that is a scalar multiple of the identity.
(ii) Deduce that .
(iii) Show that, if is faithful, then is cyclic.
[Standard results may be quoted without proof, provided they are stated clearly.]
Now let be a group of order 18 containing an elementary abelian subgroup of order 9 and an element of order 2 with for each . By considering the action of on an irreducible -module prove that has no faithful irreducible complex representation.
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Paper 2, Section II, I
2011 commentState Maschke's Theorem for finite-dimensional complex representations of the finite group . Show by means of an example that the requirement that be finite is indispensable.
Now let be a (possibly infinite) group and let be a normal subgroup of finite index in . Let be representatives of the cosets of in . Suppose that is a finite-dimensional completely reducible -module. Show that
(i) if is a -submodule of and , then the set is a -submodule of ;
(ii) if is a -submodule of , then is a -submodule of ;
(iii) is completely reducible regarded as a -module.
Hence deduce that if is an irreducible character of the finite group then all the constituents of have the same degree.
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Paper 3, Section II, I
2011 commentDefine the character of a finite group which is induced by a character of a subgroup of .
State and prove the Frobenius reciprocity formula for the characters of and of .
Now suppose that has index 2 in . An irreducible character of and an irreducible character of are said to be 'related' if
Show that each of degree is either 'monogamous' in the sense that it is related to one (of degree ), or 'bigamous' in the sense that it is related to precisely two distinct characters (of degree . Show that each is related to one bigamous , or to two monogamous characters (of the same degree).
Write down the degrees of the complex irreducible characters of the alternating group . Find the degrees of the irreducible characters of a group containing as a subgroup of index 2 , distinguishing two possible cases.
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Paper 4, Section II, I
2011 commentDefine the groups and .
Show that acts on the vector space of complex matrices of the form
by conjugation. Denote the corresponding representation of on by .
Prove the following assertions about this action:
(i) The subspace is isomorphic to .
(ii) The pairing defines a positive definite non-degenerate invariant bilinear form.
(iii) The representation maps into . [You may assume that for any compact group , and any , there is a continuous group homomorphism if and only if has an -dimensional representation over .]
Write down an orthonormal basis for and use it to show that is surjective with kernel .
Use the isomorphism to write down a list of irreducible representations of in terms of irreducibles for . [Detailed explanations are not required.]
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Paper 1, Section II, 23G
2011 commentSuppose that and are Riemann surfaces, and is a discrete subset of . For any continuous map which restricts to an analytic map of Riemann surfaces , show that is an analytic map.
Suppose that is a non-constant analytic function on a Riemann surface . Show that there is a discrete subset such that, for defines a local chart on some neighbourhood of .
Deduce that, if is a homeomorphism of Riemann surfaces and is a non-constant analytic function on for which the composite is analytic on , then is a conformal equivalence. Give an example of a pair of Riemann surfaces which are homeomorphic but not conformally equivalent.
[You may assume standard results for analytic functions on domains in the complex plane.]
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Paper 2, Section II, 23G
2011 commentLet be a lattice in generated by 1 and , where is a fixed complex number with non-zero imaginary part. Suppose that is a meromorphic function on for which the poles of are precisely the points in , and for which as . Assume moreover that determines a doubly periodic function with respect to with for all . Prove that:
(i) for all .
(ii) is doubly periodic with respect to .
(iii) If it exists, is uniquely determined by the above properties.
(iv) For some complex number satisfies the differential equation .
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Paper 3, Section II, G
2011 commentState the Classical Monodromy Theorem for analytic continuations in subdomains of the plane.
Let be positive integers with and set . By removing semi-infinite rays from , find a subdomain on which an analytic function may be defined, justifying this assertion. Describe briefly a gluing procedure which will produce the Riemann surface for the complete analytic function .
Let denote the set of th roots of unity and assume that the natural analytic covering map extends to an analytic map of Riemann surfaces , where is a compactification of and denotes the extended complex plane. Show that has precisely branch points if and only if divides .
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Paper 1, Section I, J
2011 commentLet be independent identically distributed random variables with model function , and denote by and expectation and variance under , respectively. Define . Prove that . Show moreover that if is any unbiased estimator of , then its variance satisfies . [You may use the Cauchy-Schwarz inequality without proof, and you may interchange differentiation and integration without justification if necessary.]
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Paper 2, Section I, J
2011 commentLet be a probability density function, with cumulant generating function . Define what it means for a random variable to have a model function of exponential dispersion family form, generated by . Compute the cumulant generating function of and deduce expressions for the mean and variance of that depend only on first and second derivatives of .
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Paper 3, Section I, J
2011 commentDefine a generalised linear model for a sample of independent random variables. Define further the concept of the link function. Define the binomial regression model with logistic and probit link functions. Which of these is the canonical link function?
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Paper 4, Section I, J
2011 commentThe numbers of ear infections observed among beach and non-beach (mostly pool) swimmers were recorded, along with explanatory variables: frequency, location, age, and sex. The data are aggregated by group, with a total of 24 groups defined by the explanatory variables.
The data look like this:
Let denote the expected number of ear infections of a person in group . Explain why it is reasonable to model count as Poisson with mean .
We fit the following Poisson model:
where is an offset, i.e. an explanatory variable with known coefficient produces the following (abbreviated) summary for the main effects model:

Why are expressions freq , locB, age , and sexF not listed?
Suppose that we plan to observe a group of 20 female, non-frequent, beach swimmers, aged 20-24. Give an expression (using the coefficient estimates from the model fitted above) for the expected number of ear infections in this group.
Now, suppose that we allow for interaction between variables age and sex. Give the command for fitting this model. We test for the effect of this interaction by producing the following (abbreviated) ANOVA table:

Briefly explain what test is performed, and what you would conclude from it. Does either of these models fit the data well?
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Paper 1, Section II, J
2011 commentThe data consist of the record times in 1984 for 35 Scottish hill races. The columns list the record time in minutes, the distance in miles, and the total height gained during the route. The data are displayed in as follows (abbreviated):
Consider a simple linear regression of time on dist and climb. Write down this model mathematically, and explain any assumptions that you make. How would you instruct to fit this model and assign it to a variable hills. ?
First, we test the hypothesis of no linear relationship to the variables dist and climb against the full model. provides the following ANOVA summary:

Using the information in this table, explain carefully how you would test this hypothesis. What do you conclude?
The command
summary (hills. Im1)
provides the following (slightly abbreviated) summary:

Carefully explain the information that appears in each column of the table. What are your conclusions? In particular, how would you test for the significance of the variable climb in this model?

Figure 1: Hills data: diagnostic plots
Finally, we perform model diagnostics on the full model, by looking at studentised residuals versus fitted values, and the normal QQ-plot. The plots are displayed in Figure Comment on possible sources of model misspecification. Is it possible that the problem lies with the data? If so, what do you suggest?
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Paper 4, Section II, J
2011 commentConsider the general linear model , where the matrix has full rank , and where has a multivariate normal distribution with mean zero and covariance matrix . Write down the likelihood function for and derive the maximum likelihood estimators of . Find the distribution of . Show further that and are independent.
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Paper 1, Section II, D
2011 commentDescribe the physical relevance of the microcanonical, canonical and grand canonical ensembles. Explain briefly the circumstances under which all ensembles are equivalent.
The Gibbs entropy for a probability distribution over states is
By imposing suitable constraints on , show how maximising the entropy gives rise to the probability distributions for the microcanonical and canonical ensembles.
A system consists of non-interacting particles fixed at points in a lattice. Each particle has three states with energies . If the system is at a fixed temperature , determine the average energy and the heat capacity . Evaluate each in the limits and .
Describe a configuration of the system that would have negative temperature. Does this system obey the third law of thermodynamics?
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Paper 2, Section II, D
2011 commentWrite down the partition function for a single classical non-relativistic particle of mass moving in three dimensions in a potential and in equilibrium with a heat bath at temperature .
A system of non-interacting classical non-relativistic particles, in equilibrium at temperature , is placed in a potential
where is a positive integer. Using the partition function, show that the free energy is
where
Explain the physical relevance of the constant term in the expression .
Viewing as an external parameter, akin to volume, compute the conjugate pressure and show that the equation of state coincides with that of an ideal gas.
Compute the energy , heat capacity and entropy of the gas. Determine the local particle number density as a function of .
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Paper 3, Section II, D
2011 commentA gas of non-interacting particles has energy-momentum relationship for some constants and . Determine the density of states in a threedimensional volume .
Explain why the chemical potential satisfies for the Bose-Einstein distribution.
Show that an ideal quantum Bose gas with the energy-momentum relationship above has
If the particles are bosons at fixed temperature and chemical potential , write down an expression for the number of particles that do not occupy the ground state. Use this to determine the values of for which there exists a Bose-Einstein condensate at sufficiently low temperatures.
Discuss whether a gas of photons can undergo Bose-Einstein condensation.
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Paper 4, Section II, D
2011 comment(i) Define the Gibbs free energy for a gas of particles with pressure at a temperature . Explain why it is necessarily proportional to the number of particles in the system. Given volume and chemical potential , prove that
(ii) The van der Waals equation of state is
Explain the physical significance of the terms with constants and . Sketch the isotherms of the van der Waals equation. Show that the critical point lies at
(iii) Describe the Maxwell construction to determine the condition for phase equilibrium. Hence sketch the regions of the van der Waals isotherm at that correspond to metastable and unstable states. Sketch those regions that correspond to stable liquids and stable gases.
(iv) Show that, as the critical point is approached along the co-existence curve,
Show that, as the critical point is approached along an isotherm,
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Paper 1, Section II, J
2011 commentIn a one-period market, there are assets whose prices at time are given by . The prices of the assets at time 1 have a distribution, with non-singular covariance , and the prices at time 0 are known constants. In addition, there is a bank account giving interest , so that one unit of cash invested at time 0 will be worth units of cash at time 1 .
An agent with initial wealth chooses a portfolio of the assets to hold, leaving him with in the bank account. His objective is to maximize his expected utility
Find his optimal portfolio in each of the following three situations:
(i) is unrestricted;
(ii) no investment in the bank account is allowed: ;
(iii) the initial holdings of cash must be non-negative.
For the third problem, show that the optimal initial holdings of cash will be zero if and only if
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Paper 2, Section II, J
2011 commentConsider a symmetric simple random walk taking values in statespace , where an integer . Writing , the transition probabilities are given by
where .
What does it mean to say that is a martingale? Find a condition on and such that
is a martingale. If for some real , show that is a martingale if
Suppose that the random walk starts at position at time 0 , and suppose that
Stating fully any results to which you appeal, prove that
where is as given at . Deduce that as
and comment briefly on this result.
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Paper 3, Section II, J
2011 commentFirst, what is a Brownian motion?
(i) The price of an asset evolving in continuous time is represented as
where is a standard Brownian motion, and and are constants. If riskless investment in a bank account returns a continuously-compounded rate of interest , derive a formula for the time-0 price of a European call option on the asset with strike and expiry . You may use any general results, but should state them clearly.
(ii) In the same financial market, consider now a derivative which pays
at time . Find the time-0 price for this derivative. Show that it is less than the price of the European call option which you derived in (i).
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Paper 4, Section II, J
2011 commentIn a two-period model, two agents enter a negotiation at time 0 . Agent knows that he will receive a random payment at time , where the joint distribution of is known to both agents, and . At the outcome of the negotiation, there will be an agreed risk transfer random variable which agent 1 will pay to agent 2 at time 1 . The objective of agent 1 is to maximize , and the objective of agent 2 is to maximize , where the functions are strictly increasing, strictly concave, , and have the properties that
Show that, unless there exists some such that
the risk transfer could be altered to the benefit of both agents, and so would not be the conclusion of the negotiation.
Show that, for given , the relation determines a unique risk transfer , and that is a function of .
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Paper 1, Section I, F
2011 comment(i) State the Baire Category Theorem for metric spaces in its closed sets version.
(ii) Let be a complex analytic function which is not a polynomial. Prove that there exists a point such that each coefficient of the Taylor series of at is non-zero.
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Paper 2, Section I, F
2011 comment(i) Let be any set of distinct numbers. Show that there exist numbers such that the formula
is valid for every polynomial of degree .
(ii) For , let be the Legendre polynomial, over , of degree . Suppose that are the roots of , and are the numbers corresponding to as in (i).
[You may assume without proof that for has distinct roots in ]
Prove that the integration formula in (i) is now valid for any polynomial of degree .
(iii) Is it possible to choose distinct points and corresponding numbers such that the integration formula in (i) is valid for any polynomial of degree ? Justify your answer.
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Paper 3, Section I, F
2011 commentLet .
(i) Prove that, for any with and any , there exists a complex polynomial such that
(ii) Does there exist a sequence of polynomials such that for every Justify your answer.
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Paper 4, Section I,
2011 comment(a) Let be a continuous map such that . Define the winding number of about the origin. State precisely a theorem about homotopy invariance of the winding number.
(b) Let and let be a continuous map satisfying
for each .
(i) For , let . If for each , prove that .
[Hint: Consider a suitable homotopy between and the map ,
(ii) Deduce that for some .
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Paper 2, Section II, F
2011 commentLet be the space of real continuous functions on the interval . A mapping is said to be positive if for each with , and linear if for all functions and constants .
(i) Let be a sequence of positive, linear mappings such that uniformly on for the three functions . Prove that uniformly on for every .
(ii) Define by
where . Using the result of part (i), or otherwise, prove that uniformly on .
(iii) Let and suppose that
for each Prove that must be the zero function.
[You should not use the Stone-Weierstrass theorem without proof.]
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Paper 3, Section II, F
2011 commentLet be continuous and let be a positive integer. For a continuous function, write .
(i) Let be a polynomial of degree at most with the property that there are distinct points with such that
for each . Prove that
for every polynomial of degree at most .
(ii) Prove that there exists a polynomial of degree at most such that
for every polynomial of degree at most .
[If you deduce this from a more general result about abstract normed spaces, you must prove that result.]
(iii) Let be any set of distinct points in .
(a) For , let
and . Explain why there is a unique number such that the degree of the polynomial is at .
(b) Let . Deduce from part (a) that there exists a polynomial of degree at most such that
for every polynomial of degree at most .
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Paper 1, Section II, 39B
2011 commentAn inviscid fluid with sound speed occupies the region enclosed by the rigid boundaries of a rectangular waveguide. Starting with the acoustic wave equation, find the dispersion relation for the propagation of sound waves in the -direction.
Hence find the phase speed and the group velocity of both the dispersive modes and the nondispersive mode, and sketch the form of the results for .
Define the time and cross-sectional average appropriate for a mode with frequency . For each dispersive mode, show that the average kinetic energy is equal to the average compressive energy.
A general multimode acoustic disturbance is created within the waveguide at in a region around . Explain briefly how the amplitude of the disturbance varies with time as at the moving position for each of the cases , and . [You may quote without proof any generic results from the method of stationary phase.]
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Paper 2, Section II, 38B
2011 commentA uniform elastic solid with wavespeeds and occupies the region . An -wave with displacement
is incident from on a rigid boundary at . Find the form and amplitudes of the reflected waves.
When is the reflected -wave evanescent? Show that if the -wave is evanescent then the amplitude of the reflected -wave has the same magnitude as the incident wave, and interpret this result physically.
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Paper 3, Section II, 38B
2011 commentThe dispersion relation in a stationary medium is given by , where is a known function. Show that, in the frame of reference where the medium has a uniform velocity , the dispersion relation is given by .
An aircraft flies in a straight line with constant speed through air with sound speed . If show that, in the reference frame of the aircraft, the steady waves lie behind it on a cone of semi-angle . Show further that the unsteady waves are confined to the interior of the cone.
A small insect swims with constant velocity over the surface of a pool of water. The resultant capillary waves have dispersion relation on stationary water, where and are constants. Show that, in the reference frame of the insect, steady waves have group velocity
where . Deduce that the steady wavefield extends in all directions around the insect.
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Paper 4, Section II, 38B
2011 commentShow that, in the standard notation for one-dimensional flow of a perfect gas, the Riemann invariants are constant on characteristics given by
Such a gas occupies the region in a semi-infinite tube to the right of a piston at . At time , the piston and the gas are at rest, , and the gas is uniform with . For the piston accelerates smoothly in the positive -direction. Show that, prior to the formation of a shock, the motion of the gas is given parametrically by
in a region that should be specified.
For the case , where is a constant, show that a shock first forms in the gas when
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Paper 4 , Section II, D
2011 commentDerive the Euler-Lagrange equation for the integral
where the endpoints are fixed, and and take given values at the endpoints.
Show that the only function with and as for which the integral
is stationary is .